You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, AD is parallel to BC, CD intersects AB at E, EF is parallel to BC, AE:EB=1:2, the area of triangle ADE is 1. What is the area of triangle AEF?
(A) 1/3
(B) 1/2
(C) 2/3
(D) 3/4

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, vertices A, B, and C of triangle ABC lie on circle O, with angle OAC measuring 40° and angle OBC measuring 15°. What is the degree measure of angle AOB?
Choices:
(A) 55°
(B) 110°
(C) 120°
(D) 150°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the circle O, chord BC is equal to 1. Point A is a point on the circle, and angle BAC is 30°. What is the radius of circle O?
Choices:
(A) 1
(B) 2
(C) √{3}
(D) √{5}

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x and round the side measurement to the nearest tenth.
Options:
(A) 2.0
(B) 4.1
(C) 7.9
(D) 8.1

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, points A, B, and C lie on circle O, with angle A measuring 32°. What is the degree measure of angle BOC?
Choices:
(A) 30°
(B) 64°
(C) 50°
(D) 28°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, PA is a tangent to circle O at point A, and PO intersects circle O at point C, with BC being connected. If ∠B = 20°, what is the measure of ∠P?
(A) 20°
(B) 30°
(C) 40°
(D) 50°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in quadrilateral ABCD, point E lies on the extension of side AD, connecting BE intersects side DC at point F. If the area of quadrilateral ABFD is S1 and the area of triangle CEF is S2, and the area of quadrilateral ABCD is 4, what is the value of S1 - S2?
Choices:
(A) 1
(B) 2
(C) 3
(D) 4

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, if △ABC is congruent to △AED, and ∠B is 31°, ∠C is 60°, what is the degree measure of ∠DAE?
Choices:
(A) 85°
(B) 87°
(C) 89°
(D) 91°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, PA and PB are tangents to circle O, with A and B as the points of tangency. If ∠P = 60°, PA = 2, what is the diameter of circle O?
Choices:
(A) \frac{2√{3}}{3}
(B) \frac{4√{3}}{3}
(C) 2
(D) 1

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the rhombus ABCD shown in the figure, ∠ABC = 60°, AB = 4. If points E, F, G, H are the midpoints of sides AB, BC, CD, DA respectively, and segments EF, FG, GH, and HE are connected, what is the area of quadrilateral EFGH?
Choices:
(A) 8
(B) 6√{3}
(C) 4√{3}
(D) 6

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, where AB=AC and BC=12cm, point D lies on AC with DC=4cm. If segment DC is translated 7cm along the direction of CB to form segment EF, where points E and F lie on sides AB and BC respectively, what is the perimeter of triangle EBF?
(A) 7cm
(B) 11cm
(C) 13cm
(D) 16cm

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in quadrilateral ABCD, AB = 18, AD = 12, the angle bisector of ∠ABC intersects CD at point F, and intersects the extension of AD at point E. CG is perpendicular to BE, with G as the foot of the perpendicular. If EF = 4, what is the length of segment CG?
Choices:
(A) 2√{35}
(B) 6√{3}
(C) 4√{10}
(D) 8√{2}

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, points A, B, and C lie on circle O, with angle ACB measuring 50°. What is the measure of angle AOB?
(A) 130°
(B) 100°
(C) 50°
(D) 40°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, the perpendicular bisector of AB intersects AB at E and intersects BC at D, with AD being connected. If AC is 4cm and the perimeter of triangle ADC is 11cm, what is the length of BC?
Choices:
(A) 5cm
(B) 6cm
(C) 7cm
(D) 8cm

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, the perpendicular bisector of side AB intersects AB and AC at points D and E respectively, while the perpendicular bisector of side BC intersects BC and AC at points F and G respectively. BE and BG are connected. If the perimeter of triangle BEG is 20 and GE = 2, what is the length of side AC?
Choices:
(A) 13
(B) 14
(C) 15
(D) 16

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in circle O, AB and AC are two perpendicular chords, with OD perpendicular to AB at D, and OE perpendicular to AC at E. If AB = 8cm and AC = 6cm, what is the length of the radius OA of circle O?
Choices:
(A) 4cm
(B) 5cm
(C) 6cm
(D) 8cm

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, PA, PB, and DE intersect ⊙O at A, B, and C respectively. If PA = 5cm, what is the perimeter of triangle PDE?
Choices:
(A) 5
(B) 7
(C) 8
(D) 10

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

If the radius (r) is equal to 4, what is the area of the shaded region among the given choices?
(A) 16 - 16π
(B) 16 - 8π
(C) 64 - 16π
(D) 64 - 8π

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram shown, the positions of lines a, b, c, and d are as indicated. If ∠1 = ∠2 = 90°, and ∠3 = 42°, what is the measure of ∠4?
Choices:
(A) 130°
(B) 138°
(C) 140°
(D) 142°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, it is known that AD is the angle bisector of triangle ABC, DE is parallel to AB and intersects AC at E. If AE/EC = 3/5, what is the value of AC/AB?
Choices:
(A) 3/5
(B) 3/2
(C) 8/5
(D) 5/3

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, points A, B, and C are all on circle O. If angle A is 64°, what is the degree measure of angle OCB?
Choices:
(A) 24°
(B) 26°
(C) 28°
(D) 30°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, one edge of the ruler passes through a right angle vertex, and the other edge of the ruler intersects with one side of the right angle. If ∠1 = 30°, what is the degree measure of ∠2?
Choices:
(A) 30°
(B) 45°
(C) 60°
(D) 120°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In circle A with a radius of 5, chords BC and ED are subtended by central angles ∠BAC and ∠EAD respectively. Given that DE = 6, ∠BAC + ∠EAD = 180°, what is the length of chord BC?
Choices:
(A) √
(B) √
(C) 8
(D) 6

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that triangle PQS is congruent to triangle RQS, what is the value of x?
Options:
(A) 8
(B) 12
(C) 16
(D) 20

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, if ∠AOB = 20°, ∠AOC = 90°, and points B, O, D are collinear, what is the measure of ∠COD in degrees?
Choices:
(A) 100°
(B) 105°
(C) 110°
(D) 115°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, point D is any point on side BC of triangle ABC, points E and F are the midpoints of segments AD and CE, respectively. If the area of triangle ABC is 1, what is the area of triangle BEF?
Choices:
(A) 0.5
(B) 1/3
(C) 1/4
(D) 2/5

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, quadrilateral ABCD is a rhombus with side length 5cm. The diagonals BD and AC intersect at point O, with BD = 6cm. What is the length of diagonal AC?
Choices:
(A) 8cm
(B) 4cm
(C) 3cm
(D) 6cm

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, it is known that line BC is tangent to circle O at point C, PD is the diameter of circle O, the extension of line BP intersects the extension of line CD at point A, angle A is 28°, and angle B is 26°. What is the measure of angle PDC?
(A) 34°
(B) 36°
(C) 38°
(D) 40°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, triangles ABC and DEC are congruent. A perpendicular line AF is drawn from point A to CD, with F as the foot of the perpendicular. If angle BCE is 65°, what is the degree measure of angle CAF?
Choices:
(A) 15°
(B) 25°
(C) 35°
(D) 65°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, points C and D are on circle O, and A is the midpoint of arc DC. If ∠ABD = 15°, what is the degree measure of ∠BOC?
Choices:
(A) 120°
(B) 150°
(C) 210°
(D) 75°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, A, B, and C are three points on circle O. If ∠AOB is 60° and ∠B is 55°, what is the degree measure of ∠A?
Choices:
(A) 25°
(B) 30°
(C) 40°
(D) 50°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, line AB is tangent to circle O at point B. The extension of line AO intersects circle O at point C, forming line BC. If angle A measures 42°, what is the measure of angle C?
Choices:
(A) 30°
(B) 42°
(C) 48°
(D) 24°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the measure of angle 2.
Options:
(A) 59
(B) 74
(C) 106
(D) 110

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, the radius of circle O, OC, is 10cm. Line AB is perpendicular to OC, with the foot of the perpendicular being H, intersecting circle O at points A and B. If AB measures 12cm, then when line AB is translated downwards along OC to be tangent to circle O, what is the distance of the translation?
Choices:
(A) 2cm
(B) 3cm
(C) 4cm
(D) 5cm

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, AB is the diameter of circle O, with points C and D on the circle. If AC, BC, AD, and CD are connected, and ∠BAC = 40°, what is the measure of ∠ADC?
Choices:
(A) 30°
(B) 40°
(C) 50°
(D) 60°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, triangle ABC is inscribed in circle O, where OD is perpendicular to BC at D and angle DOC is 50°. What is the degree measure of angle A?
Choices:
(A) 40°
(B) 45°
(C) 50°
(D) 60°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, with AD as the angle bisector of angle BAC, DE perpendicular to AB with E as the foot of the perpendicular, if AB is 10cm and AC is 6cm, what is the length of BE?
Choices:
(A) 10cm
(B) 6cm
(C) 4cm
(D) 2cm

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that the measure of angle 11 is 62 degrees and angle 14 is 38 degrees in the figure, what is the measure of angle 8?
Choices:
(A) 38
(B) 62
(C) 118
(D) 142

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, lines AB and CD intersect at point O, with OA bisecting angle EOC. If angle BOD is 42°, what is the degree measure of angle EOD?
Choices:
(A) 96°
(B) 94°
(C) 104°
(D) 106°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the similar triangles △ABC and △DEF, with a similarity ratio of 1:2, if DE = 4, what is the length of AB?
Choices:
(A) 2
(B) 4
(C) 8
(D) 1

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate the perimeter of the figure in feet and round to the nearest tenth, if needed.
Choices:
(A) 20
(B) 28
(C) 36
(D) 44

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in quadrilateral ABCD, AE is perpendicular to BC with E as the foot of the perpendicular, and AF is perpendicular to CD with F as the foot of the perpendicular. If AE:AF = 2:3 and the perimeter of ABCD is 10, what is the length of AB?
Choices:
(A) 2
(B) 2.5
(C) 3
(D) 3.5

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, where angle ACB is 90° and angle B is 40°, arcs are drawn with points A and B as centers and the same length (greater than 0.5 times AB) as radii. These arcs intersect at points M and N. A line is drawn MN intersecting AB at point D and BC at point E. What is the measure of angle CDE?
Choices:
(A) 8°
(B) 10°
(C) 15°
(D) 20°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, triangle ABC has side AC intersecting circle O at points C and D, passing through the center O, and side AB tangent to circle O at point B. If angle A is 30°, what is the measure of angle C?
Choices:
(A) 40°
(B) 30°
(C) 60°
(D) 45°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, DE is parallel to BC, and DE intersects AB and AC at points D and E. If AD = 4 and DB = 2, what is the ratio of DE to BC?
Choices:
(A) 2/3
(B) 1/2
(C) 3/4
(D) 3/5

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, the diagonals AC and BD of rhombus ABCD intersect at point O. Point D is perpendicular to line AB at point H, and line OH is drawn. If OA is 6 and the area of rhombus ABCD is 48, what is the length of OH?
Choices:
(A) 4
(B) 8
(C) √{13}
(D) 6

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the equilateral triangle △ABC shown in the figure, D is a point on side BC, E is a point on side AC, and ∠ADB + ∠EDC = 120°. If BD = 3 and CE = 2, what is the length of the sides of △ABC?
Choices:
(A) 9
(B) 12
(C) 16
(D) 18

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, C is a point on circle O, AD is perpendicular to the tangent passing through point C, with the foot of the perpendicular being D. If ∠DCA = 55°, what is the degree measure of ∠CAO?
Choices:
(A) 25°
(B) 35°
(C) 45°
(D) 55°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, points A, B, and C are all on circle O, with point C on the arc of AB. If ∠AOB = 64°, what is the degree measure of ∠ACB?
Choices:
(A) 26°
(B) 30°
(C) 32°
(D) 64°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the parallelogram ABCD shown, where ∠B = 60°, AB = 4, BC = 5, P is any point on diagonal AC (not coinciding with points A or C), and PE is parallel to BC intersecting AB at E, and PF is parallel to CD intersecting AD at F, what is the area of the shaded region?
Choices:
(A) 5
(B) 5√{3}
(C) 10
(D) 10√{3}

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, AB is the diameter of circle O, and points C and D are on circle O. If ∠CAB = 25°, what is the measure of angle D?
Choices:
(A) 85°
(B) 105°
(C) 115°
(D) 130°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, points A, B, and C lie on circle O, with angle BOC measuring 72°. What is the measure of angle BAC?
(A) 36°
(B) 26°
(C) 72°
(D) 108°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, if the diameter AB of circle O forms an angle of 35° with chord AC, and the tangent PC passing through point C intersects the extension of AB at point P, what is the measure of angle ∠P?
Choices:
(A) 15°
(B) 20°
(C) 25°
(D) 30°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, where AB = AC = 12, BC = 10, AD bisects angle BAC and intersects BC at point D, and E is the midpoint of AC. If DE is connected, what is the perimeter of triangle CDE?
Choices:
(A) 11
(B) 17
(C) 18
(D) 16

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, given line segment AB, a perpendicular line is drawn from point B on AB, and a point C is taken on the perpendicular line such that BC = 1/2 AB; line AC is drawn, with point C as the center, an arc is drawn with CB as the radius intersecting AC at point D; then, with point A as the center, an arc is drawn with AD as the radius intersecting AB at point P, what is the value of AP/AB?
Choices:
(A) (sqrt(5)-1)/2
(B) (sqrt(5)+1)/2
(C) (3-sqrt(5))/2
(D) sqrt(2)/2

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given figure, AB is the diameter of a semicircle, O is the center of the circle, C is a point on the semicircle, and D is a point on line AC. If ∠BOC = 50°, what is the measure of angle D?
Choices:
(A) 105°
(B) 115°
(C) 125°
(D) 85°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, if the right angle of the triangle is placed on one side of the ruler and ∠1 is 65°, what is the degree measure of ∠2?
Choices:
(A) 10°
(B) 15°
(C) 20°
(D) 25°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in circle O, if chord AB is parallel to CD and angle AOC is 70°, what is the degree measure of angle BAD?
Choices:
(A) 30°
(B) 35°
(C) 70°
(D) 140°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x that would make the quadrilateral a parallelogram.
Choices:
(A) 6
(B) 12
(C) 24
(D) 30

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, point D is on side AB, and DE is parallel to BC intersecting AC at point E. If AD/DB = 2/3 and AE = 6, what is the length of EC?
(A) 6
(B) 9
(C) 15
(D) 18

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the values of x that would make the quadrilateral a parallelogram.
Choices:
(A) 4
(B) 6
(C) 8
(D) 16

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, points A, B, C, and P lie on circle O. CD is perpendicular to OA, CE is perpendicular to OB, with D and E as the respective foot of the perpendiculars. If angle DCE is 80°, what is the degree measure of angle P?
Choices:
(A) 100°
(B) 80°
(C) 60°
(D) 50°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of y in the given figure. If the measure of angle 3 is represented as 4y + 30 and the measure of angle 7 is represented as 7y + 6, what is the value of y?
Choices:
(A) 8
(B) 9
(C) 10
(D) 12

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, a square BCDE is constructed inside the regular hexagon ABCDEF, with the diagonal AH. What is the measure of angle HAB?
Choices:
(A) 75°
(B) 60°
(C) 55°
(D) 45°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, triangle ABC is inscribed in circle O, AD is the diameter of circle O, intersecting BC at point E. If DE = 2 and OE = 3, what is the value of tanC * tanB?
(A) 2
(B) 3
(C) 4
(D) 5

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a rhombus ABCD, if BD = 8 and the area is 24, what is the perimeter of rhombus ABCD?
Choices:
(A) 5
(B) 10
(C) 10√{2}
(D) 20

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

If D A = 7.4 in circle F, what is the value of E F?
Choices:
(A) 3.7
(B) 7.4
(C) 11.1
(D) 14.8

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, ∠MON = 90°. Points A and B are moving along the rays OM and ON respectively. BE bisects ∠NBA, and the extension of BE intersects the bisector of ∠BAO at point C. What is the measure of ∠C?
Choices:
(A) 30°
(B) 45°
(C) 55°
(D) 60°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, to measure the distance between two points A and B located next to a pond, point O is marked on the ground. Midpoints C and D are determined on OA and OB respectively, and CD is measured to be 10m. What is the distance between points A and B?
Choices:
(A) 5m
(B) 10m
(C) 20m
(D) 40m

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, as shown in the diagram, DE is parallel to BC, D is on AB, E is on AC, and DF is parallel to AC intersecting BC at point F. If AE = 5, EC = 3, and BF = 1.5, what is the length of BC?
(A) 2.5
(B) 4
(C) 3
(D) 5

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, the base points A and B of a protractor slide along the positive y-axis and negative x-axis respectively. Point D is located at the 128° mark on the protractor. When the distance between point D and the origin O is maximum, what is the measure of angle OAB?
Choices:
(A) 64°
(B) 52°
(C) 38°
(D) 26°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram below, AB is parallel to EF which is parallel to CD. If ∠ABC is 46° and ∠BCE is 20°, what is the measure of ∠CEF?
Choices:
(A) 144°
(B) 154°
(C) 164°
(D) 160°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, quadrilateral ABCD is inscribed in circle O. If ∠BOD = 100°, what is the degree measure of ∠DCB?
Choices:
(A) 50°
(B) 80°
(C) 100°
(D) 130°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, points C and D are on circle O, and ∠ABC = 50°. What is the measure of angle D?
Choices:
(A) 30°
(B) 40°
(C) 50°
(D) 60°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the perimeter of triangle DEF, given that triangle DEF is similar to triangle CBF, the perimeter of triangle CBF is 27, DF is 6, and FC is 8.
Choices:
(A) 16
(B) 17.25
(C) 20.25
(D) 27

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, the zero degree lines of two protractors are both on the line AB, and the center of the smaller protractor is on the outer edge of the larger protractor. If the point P on their outer edge corresponds to 50° on the larger protractor, what is the degree measure of ∠PBA?
(A) 30°
(B) 32.5°
(C) 35°
(D) 37.5°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, ∠AOB = 30°, point P is a point inside ∠AOB, OP = 10, points M and N are on OA and OB respectively. Find the minimum perimeter of triangle PMN.
Choices:
(A) 5
(B) 10
(C) 15
(D) 20

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, if it is known that a is parallel to b and angle 1 is 73°, what is the measure of angle 2?
Choices:
(A) 73°
(B) 97°
(C) 107°
(D) 117°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, point C lies on circle O, and ∠B is 65°. What is the measure of angle A?
Choices:
(A) 20°
(B) 25°
(C) 30°
(D) 35°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, if a right-angled triangle ruler with a 30° angle is placed with its right angle vertex on one side of the ruler, and if ∠1 = 20°, what is the degree measure of ∠2?
Choices:
(A) 20°
(B) 50°
(C) 60°
(D) 70°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, triangle ABC is inscribed in circle O, AD is perpendicular to BC, BE is perpendicular to AC, and AD and BE intersect at point M. If AC = 8 and BM = 4, what is the radius of circle O?
Choices:
(A) 2√{5}
(B) 2√{3}
(C) 4√{2}
(D) 6

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, D, E, F are the midpoints of sides BC, AC, AB respectively. AH is perpendicular to BC at H, FD = 16. What is the length of HE?
Choices:
(A) 32
(B) 16
(C) 8
(D) 10

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, the tangent passing through point A on circle O intersects the extension of chord BC at point P. If ∠CAP = 40° and ∠ACP = 100°, what is the degree measure of ∠BAC?
Choices:
(A) 40°
(B) 60°
(C) 80°
(D) 70°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, and ∠ABC is 40°. What is the measure of ∠BAC?
(A) 90°
(B) 50°
(C) 60°
(D) 65°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that P Q is congruent to Q S, Q R is congruent to R S, and the measure of angle P R S is 72 degrees, what is the measure of angle Q P S?
Choices:
(A) 27
(B) 54
(C) 63
(D) 72

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, BD and CE are the angle bisectors of triangle ABC. AN is perpendicular to BD at point N, and AM is perpendicular to CE at point M. If the perimeter of triangle ABC is 17 and BC is 7, what is the length of MN?
Choices:
(A) \frac{3}{2}
(B) 2
(C) \frac{5}{2}
(D) 3

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the rhombus ABCD, if the diagonal AC is 8 and BD is 10, what is the area of triangle AOD?
Choices:
(A) 9
(B) 10
(C) 11
(D) 12

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, if △ABC is congruent to △ADC, ∠ABC is 118°, and ∠DAC is 40°, what is the degree measure of ∠BCD?
Choices:
(A) 40°
(B) 44°
(C) 50°
(D) 84°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, if angle A is 66° and point I is the incenter, what is the measure of angle BIC?
Choices:
(A) 114°
(B) 122°
(C) 123°
(D) 132°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, given that AB is parallel to CD and EF, AC = 4, CE = 1, BD = 3, what is the value of DF?
Choices:
(A) 1/2
(B) 4/3
(C) 3/4
(D) 1

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the right triangle ABC, where angle ACB is 90 degrees, squares are constructed outwardly on each side of the triangle. The areas of these squares are denoted as S1, S2, and S3, with S1 = 4 and S2 = 16. What is the length of side AB in triangle ABC?
Choices:
(A) 2√{3}
(B) 12
(C) 2√{5}
(D) 20

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that the vertices of quadrilateral ABCD intersect square EFGH and divide its sides into segments with a ratio of 1:2, what is the ratio between the areas of ABCD and EFGH?
Choices:
(A) 1:4
(B) 1:2
(C) 5:9
(D) 1: √2

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In rhombus ABCD, if EB = 9, AB = 12, and angle ABD measures 55 degrees, what is the length of CE?
Choices:
(A) \sqrt { 63 }
(B) \sqrt { 70 }
(C) 9
(D) 12

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, AB is parallel to CD and EF, ∠ABE = 70°, ∠DCE = 144°, what is the degree measure of ∠BEC?
Choices:
(A) 34°
(B) 36°
(C) 44°
(D) 46°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x.
Options:
(A) 7
(B) 7 \sqrt { 3 }
(C) 14
(D) 7 \sqrt { 5 }

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate the area of the figure and round to the nearest tenth if needed.
Choices:
(A) 36
(B) 120
(C) 156
(D) 192

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, the diameter of the protractor coincides with the hypotenuse of the right triangle ABC, where the endpoint N of the protractor's 0-degree line coincides with point A. Ray CP rotates around point C, starting from CA in a clockwise direction at a speed of 2° per second. Point E is the intersection of CP and the semicircular arc of the protractor. At the 35th second, what is the reading corresponding to point E on the protractor?
Choices:
(A) 35°
(B) 70°
(C) 100°
(D) 140°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, it is known that line a is parallel to line b, line c intersects both lines a and b, and angle 1 is 60°. What is the measure of angle 2?
Choices:
(A) 60°
(B) 70°
(C) 120°
(D) 150°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is a chord of circle O, OD is perpendicular to AB with foot C, intersecting circle O at point D, and point E lies on the major arc AB. If ∠AOD = 52°, what is the degree measure of ∠DEB?
Choices:
(A) 52°
(B) 40°
(C) 26°
(D) 45°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, it is known that there is a small square EFGH inside the square ABCD with a side length of 12. If BF is 3, what is the length of BE?
Choices:
(A) √{12}
(B) \frac{15}{4}
(C) 5
(D) \frac{9}{4}

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, points D, E, and F are the midpoints of sides BC, AD, and CE respectively. If the area of triangle ABC is 16cm2, what is the area of the shaded region?
Choices:
(A) 8cm2
(B) 4cm2
(C) 2cm2
(D) 1cm2

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in the circumcircle of triangle ABC inscribed in circle O, angle ABC is 30°. The extension of AC intersects the tangent to circle O passing through point B at point D. If the radius of circle O, OC, is 2 and BD is parallel to OC, what is the length of CD?
Choices:
(A) 1+√3/3
(B) 4√3/3
(C) 2√3/3
(D) 2

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x.
Options:
(A) 107
(B) 117
(C) 118
(D) 135

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, where angle A is 90°, point D lies on side AC, and DE is parallel to BC. If angle 1 is 145°, what is the measure of angle B?
Choices:
(A) 35°
(B) 25°
(C) 45°
(D) 55°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, if AB is the diameter of circle O, CD is a chord of circle O, and ∠ABD = 56°, what is the measure of ∠BCD?
Choices:
(A) 34°
(B) 44°
(C) 54°
(D) 56°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, AB is the diameter of circle O, points C and D are on circle O, and points C and D are on opposite sides of AB. AD, OD, and OC are connected, if ∠AOC = 70°, and AD is parallel to OC, what is the degree measure of ∠AOD?
Choices:
(A) 70°
(B) 60°
(C) 50°
(D) 40°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In quadrilateral ABCD inscribed in a circle, if angle B is 60°, what is the measure of angle D?
Choices:
(A) 30°
(B) 40°
(C) 60°
(D) 120°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, if the degree measure of ∠AOB is 100°, what is the degree measure of ∠ACB?
Choices:
(A) 110°
(B) 120°
(C) 130°
(D) 140°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, if the side length of rhombus ABCD is 6 and ∠ABC = 60°, what is the length of diagonal AC?
Choices:
(A) 3
(B) 3√{3}
(C) 6
(D) 6√{3}

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, line AB is tangent to circle O at point B. The extension of line AO intersects circle O at point C. If angle A is 36°, what is the measure of angle C?
Choices:
(A) 36°
(B) 54°
(C) 60°
(D) 27°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x.
Options:
(A) 4
(B) 6
(C) 4 \sqrt { 3 }
(D) 8 \sqrt { 3 }

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x.
Options:
(A) 3
(B) 4
(C) 5
(D) 6

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given figure, with line a parallel to line b, angle 1 measures 60° and angle 2 measures 40°. What is the measure of angle 3?
Choices:
(A) 40°
(B) 60°
(C) 80°
(D) 100°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, angle ABC is 50° and angle ACB is 80°. If BP bisects angle ABC and CP bisects angle ACB, what is the measure of angle BPC?
(A) 100°
(B) 110°
(C) 115°
(D) 120°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, △ABC is similar to △A1B1C1 with similarity center O. If OA:OA1 = 1:2, what is the ratio of the perimeters of △ABC to △A1B1C1?
Choices:
(A) 1:2
(B) 1:3
(C) 1:4
(D) 1:√{2}

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, the height of lamp post AB is 8 meters, and the horizontal distance between tree CD and the lamp post is 4 meters. If the length of the shadow DE of the tree under the light is 3 meters, what is the height of the tree?
Choices:
(A) 4 meters
(B) 6 meters
(C) 32/3 meters
(D) 24/7 meters

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate the perimeter of the triangle and choose the correct option:
(A) 87
(B) 90
(C) 95
(D) 100.4

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the rectangle ABCD, where AB=2 and BC=3, a circle with radius 1 is drawn with center A. E is a moving point on the circle and P is a moving point on BC. What is the minimum value of PE+PD?
Choices:
(A) 2
(B) 3
(C) 4
(D) 2√{3}

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, AB and CD are both chords of circle O, and AB is perpendicular to CD. If ∠CDB = 57°, what is the degree measure of ∠ACD?
Choices:
(A) 33°
(B) 34°
(C) 43°
(D) 57°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x.
Options:
(A) 1
(B) 3
(C) 5
(D) 7

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, and C, D are two points on circle O. If ∠BCD = 40°, what is the measure of ∠ABD?
Choices:
(A) 20°
(B) 40°
(C) 50°
(D) 60°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In parallelogram ABCD, diagonals AC and BD intersect at point O, with E being the midpoint of AD and OE = 3. What is the length of CD?
Choices:
(A) 3
(B) 5
(C) 6
(D) 9

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, a semicircle with a diameter of 6 rotates clockwise 45° around the endpoint A of diameter AB. The corresponding point to B is B′. What is the area of the shaded region in the figure?
Choices:
(A) \frac{3}{4}π
(B) \frac{9}{4}π
(C) \frac{9}{2}π
(D) 9π

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, CA is a tangent to circle O at point A, and point B lies on circle O. If ∠CAB = 55°, what is the measure of ∠AOB?
Choices:
(A) 120°
(B) 110°
(C) 90°
(D) 55°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, Kongming designed a plan to measure the height of a certain ancient city wall using a flashlight. A horizontal plane mirror is placed at point P, and the light rays starting from point A reflect off the mirror and just reach the top point C of the ancient city wall CD. It is known that AB is perpendicular to BD, CD is perpendicular to BD, and AB measures 6 meters, BP measures 9 meters, and PD measures 15 meters. What is the height of the ancient city wall?
Choices:
(A) 6 meters
(B) 8 meters
(C) 10 meters
(D) 15 meters

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, E is a point on the diagonal BD of the parallelogram ABCD. Line AE is extended to intersect BC at point F, and \frac{BF}{BC}=\frac{1}{3}. What is the value of \frac{BE}{DE}?
Choices:
(A) \frac{1}{3}
(B) \frac{1}{2}
(C) \frac{2}{3}
(D) \frac{3}{4}

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, where AB = 10 and AC = 6, AD and AE are the angle bisector and median, respectively. If point C is perpendicular to AD at point F, and EF is connected, what is the length of segment EF?
Choices:
(A) 0.5
(B) 2
(C) 3.5
(D) 3

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, AB is parallel to CD, AD equals CD, and angle 1 is 65°. What is the degree measure of angle 2?
Choices:
(A) 50°
(B) 60°
(C) 65°
(D) 70°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, point D is a point on AC, and point O is a point on the side AB with AD=DO. A circle is drawn with center O and radius OD, intersecting AC at another point E and intersecting AB at points F and G. If ∠BAC=22°, what is the measure of ∠EFG?
(A) 55°
(B) 44°
(C) 38°
(D) 33°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given figure, in circle O, if ∠BOC is 50°, what is the degree measure of ∠A?
Choices:
(A) 15°
(B) 25°
(C) 50°
(D) 100°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, points A, B, and C are all on circle O, with ∠A = ∠B = 20°. What is the measure of ∠AOB?
Choices:
(A) 40°
(B) 60°
(C) 80°
(D) 100°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, triangle ABC is inscribed in circle O, with AD as the diameter of circle O. If angle C is 63°, what is the measure of angle DAB?
(A) 27°
(B) 31.5°
(C) 37°
(D) 63°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, line PA is tangent to circle O at point A, and PO intersects circle O at point C, with BC being connected. If ∠P = 40°, what is the degree measure of ∠ABC?
Choices:
(A) 20°
(B) 25°
(C) 40°
(D) 50°

Answer: $\boxed{B}$