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['The diagram consists of a circle labeled as circle O. Inside the circle, there is a square ABCD. Point P lies on the circumference of the circle and is connected to points A, B, C, and D.'] then, please answer, In the given diagram, square ABCD is enclosed within circle O, and point P is located on the side AB. What is the measure of angle BPC? | <answer> 45° </answer> |
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['The diagram consists of a triangle ABC with vertices A, B, and C. Point D is located on the circumference of the circumcircle of triangle ABC. The line segment AD intersects the extension of BC at point E.'] then, please answer, As shown in the figure, AB is the diameter of the semicircle O. C is a point on the semicircle O different from A and B. D is the midpoint of AC. Extend DC to intersect the extension line of AB at point E. If angle CAE = 14°, then the degree of angle E is () | <answer> 24° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, another line is drawn from point A to point D, which lies on the circumference of the circle.'] then, please answer, In the given diagram of circle O, where AB is the diameter and CD is a chord, if angle ACD measures 40°, what is the degree measure of angle BAD? | <answer> 50° </answer> |
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['The diagram consists of a circle denoted as circle O. On the circumference of this circle, there are three points labeled A, B, and C. The line segment AB is a diameter of the circle. Point C lies on the circumference of the circle.'] then, please answer, As shown in the figure, A, B, C are three points on circle O. angle AOB = 60°, angle B = 55°, then the degree of angle A is () | <answer> 25° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the line segment AB.'] then, please answer, Given that AB is the diameter of circle O and points C and D lie on circle O, with angle DOB measuring 140 degrees, what is the measure of angle ACD? | <answer> 20° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled A, B, and C. Point D is located on the line segment AB.'] then, please answer, As shown in the diagram, AB is the diameter of circle O, and C, D are two points on the circle. It is given that angle DCB = 30°. Thus, angle AOD = (). | <answer> 120° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point labeled as A. A line segment AB is drawn from point A to point B, which is tangent to the circle at point B. Another line segment AC is drawn from point A to point C, intersecting the circle at point C.'] then, please answer, In the given illustration, circle O has diameter AB, and chords AC and BC. If angle A measures 20°, what is the measure of angle B? | <answer> 70° </answer> |
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['The diagram consists of a circle denoted as circle O. On the circumference of this circle, there are three points labeled A, B, and C.'] then, please answer, On the depicted diagram, points A, B, and C lie on circle O. If the measure of angle ACB is 38 degrees, what is the measure of angle AOB? | <answer> 76° </answer> |
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['The diagram consists of a triangle ABC with vertices A, B, and C. Point O is located inside the triangle.'] then, please answer, If point O is the circumcenter of triangle ABC and angle A measures 40 degrees, what is the measure of angle BOC when BO and CO are connected? | <answer> 80° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled as A, B, C, and D in a clockwise order.'] then, please answer, If quadrilateral ABCD is inscribed in circle O and angle A measures 80 degrees, what is the degree measure of angle C? | <answer> 100° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Additionally, there is another point labeled as D on the circumference of the circle.'] then, please answer, As shown in the diagram, triangle ABC is inscribed in circle O, CD is the diameter of circle O, and angle ABC=20°. Then the degree of angle ACD is (). | <answer> 70° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the arc BC.'] then, please answer, As shown in the figure, circle O is the circumcircle of equilateral triangle triangle ABC. Point D is a point on arc BC, and angle CAD = 20°. The measure of angle ACD is (). | <answer> 100° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, there is a line segment connecting points B and C. The center of the circle is marked as point O.'] then, please answer, If circle O is the circumcircle of triangle ABC and angle A measures 50 degrees, and E is the midpoint of side BC, what is the measure of angle CBD when OE is extended to intersect circle O at point D and BD is connected? | <answer> 25° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled as A, B, C, and D. Point B is connected to points A and C by two line segments. Point D is also connected to point B by a line segment.'] then, please answer, In the given diagram, circle O has diameter AB, and points C and D lie on circle O. If angle BCD measures 44°, what is the degree measure of angle ABD? | <answer> 46° </answer> |
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['The diagram consists of a circle denoted as circle O. A line segment AB is drawn as the diameter of the circle, passing through the center O. Point C lies on the circumference of the circle. Additionally, there is a line segment OC connecting point O to point C.'] then, please answer, In the given diagram, if chord AB intersects radius OC at point D in circle O and BC equals CD, and AC is connected, what is the measure of angle BAC if angle B measures 52 degrees? | <answer> 14° </answer> |
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['The diagram consists of a circle denoted as circle O. On the circumference of this circle, there are three points labeled A, B, and C. The line segment AB is a diameter of the circle. Additionally, there is an angle labeled ABC.'] then, please answer, What is the degree measure of angle OCB if points A, B, and C all lie on circle O and angle A is 64° in the given diagram? | <answer> 26° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points named A, B, and C. Point D is connected to points A and B, while point E is connected to points A and C. Additionally, there is a line segment connecting points A and E.'] then, please answer, In the given diagram, circle O has diameter AB, where point C lies on AB and point D is on the circle. If AD is equal to AC, and DC is extended to intersect circle O at point E, what is the measure of angle COE if angle BAD is 30 degrees? | <answer> 30° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point labeled as A. From point A, two lines are drawn to points B and C on the circumference. Point D is located on the line segment BC.'] then, please answer, In the given diagram, the line AB is the diameter of circle O. Point C lies on the extension of line AB. A tangent CD is drawn from point C to circle O, with point D as the point of tangency. If angle ADC measures 115°, what is the degree measure of angle ACD? | <answer> 40° </answer> |
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['The diagram consists of a triangle ABC with vertices A, B, and C. Point D is located on the extension of side BC beyond point B. The line segment AD intersects the circumcircle of triangle ABC at point E. Additionally, there is a line segment BE that connects points B and E.'] then, please answer, What is the measure of angle CDB in circle O if AB is the diameter, AT is the tangent, angle T measures 40 degrees, BT intersects circle O at point C, and E is a point on AB, with the extension of CE intersecting circle O at point D? | <answer> 40° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point E is also on the circumference of the circle.'] then, please answer, As shown in the diagram, AB is the diameter of circle O. EF and EB are chords of circle O. Connecting OF, if angle AOF = 40°, then the degree of angle E is (). | <answer> 70° </answer> |
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['The diagram consists of a circle labeled as circle O. Point P is located outside the circle, and point A lies on the circumference of the circle. Line PA intersects the circle at points B and C.'] then, please answer, In the given diagram, the line segment AB is the diameter of circle O. A tangent PA is drawn from point P to circle O at A. Additionally, the line segment BC is parallel to OP and intersects circle O at C. If the measure of angle B is 70 degrees, what is the measure of angle OPC? | <answer> 20° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point labeled as A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, there is a line segment connecting points B and C.'] then, please answer, In the given diagram of triangle ABC, where AB is equal to AC and circle O is the circumcircle of triangle ABC, if the extension of BO intersects side AC at point D and angle ACB measures 70 degrees, what is the measure of angle ABD? | <answer> 20° </answer> |
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['The diagram consists of a circle labeled as circle O. A line segment AB is drawn as the diameter of the circle, passing through the center O. Point C lies on the circumference of the circle. Another line segment AC connects point A to point C. Point D is located outside the circle and is connected to points A and C by line segments AD and CD respectively.'] then, please answer, What is the measure of angle BAC if AB is the diameter of circle O, C is a point on the circle, and the tangent passing through point C intersects with the extension of diameter AB at point D, with angle ADC measuring 20°? | <answer> 35° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are six points labeled A, B, C, D, E, and F in a clockwise direction.'] then, please answer, What is the measure of angle APB if point P lies on the circumcircle of regular hexagon ABCDEF, which is circle O, and P is not coincident with A or B? | <answer> 30° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point labeled as A. From point A, two lines are drawn: one line extends to point B on the circumference and another line extends to an external point labeled as P.'] then, please answer, In the given diagram, circle O is intersected by the extension of line segment PO at point B, and PA serves as a tangent at point A. If the measure of angle BAP is 116 degrees, what is the measure of angle P? | <answer> 38° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the line segment AB.'] then, please answer, In the given illustration, AB and AC serve as tangents to circle O, intersecting at points B and C. Point D lies on circle O, and DC and BD are connected. If angle A measures 50°, what is the measure of angle D? | <answer> 65° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, there is another line connecting points B and C.'] then, please answer, What is the value of angle ADC if it is given that BC is the diameter of circle O, and points A and D lie on circle O, and angle ACB measures 32°? | <answer> 58° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the arc AB.'] then, please answer, As shown in the figure, AB is the diameter of circle O. C is a point on circle O, and D is the midpoint of BC. Connecting OC and AD, if angle AOC=60°, then the degree of angle BAD is (). | <answer> 30° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled A, B, and C. Point D is located on the extension of line segment AB.'] then, please answer, What is the measure of angle C in triangle ABC, where point O lies on side BC, and the circle with center O and radius BO intersects side AC at point A, and point D lies on the major arc AB with angle ADB measuring 60 degrees? | <answer> 30° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled A, B, and C. Point D is also on the circumference of the circle.'] then, please answer, In the given diagram, quadrilateral ABCD is inscribed in circle O. If angle BOD measures 140°, what is the degree measure of angle BCD? | <answer> 110° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point labeled as C. A line segment AB is drawn tangent to the circle at point C. Point A lies on the extended line of OC.'] then, please answer, In the given illustration, AC is the tangent line to circle O at point C. If the extended line AO intersects circle O at point B and the length of AO is 4, what can you deduce from connecting BC? | <answer> 30° </answer> |
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['The diagram consists of a circle denoted as circle O. On the circumference of the circle, there is a point labeled A. From point A, two lines are drawn: one line extends to point P and another line extends to point B. Point C lies on the circumference of the circle and is connected to points A and B.'] then, please answer, In the given diagram, tangents PA and PB intersect at circle O, with A and B as the points of tangency. If point C is situated on circle O and angle ACB measures 52°, what is the measure of angle APB? | <answer> 76° </answer> |
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['The diagram consists of a circle labeled as circle O. A line segment AB is drawn as the diameter of the circle, passing through the center O. Point C lies on the circumference of the circle.'] then, please answer, In the given diagram, if AB is a chord of circle O and AC is tangent to circle O at point A, and angle O measures 120 degrees, what is the degree measure of angle BAC? | <answer> 60° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the extended line of BC.'] then, please answer, The point E lies on the extension of line DC. If the measure of angle BCE is 105°, what is the value of angle BOD? | <answer> 150° </answer> |
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['The diagram consists of a circle denoted as circle O. On the circumference of the circle, there are three points labeled A, B, and C. Point D is also on the circumference of the circle.'] then, please answer, If CD is the diameter of circle O and points A and B lie on circle O such that AC = AB, and angle BCD measures 26°, what is the degree measure of angle ABC? | <answer> 32° </answer> |
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['The diagram consists of a triangle ABC with vertices A, B, and C. Point D is located on side AB, and point E is located on side AC. Line DE is parallel to side BC.'] then, please answer, In the given diagram of triangle ABC, points D and E are located on sides AB and AC respectively. It is known that DE is parallel to BC. If AD measures 2.0 units, DB measures 1.0 unit, and DE measures 1.6 units, what is the length of BC? | <answer> 2.4cm </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, another line is drawn from point A to point D on the circumference of the circle.'] then, please answer, If AB is the diameter of circle O, and C and D are points on circle O with angle CDB measuring 20 degrees, then what is the measure of angle ABC? | <answer> 70° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two tangents are drawn to the circle, intersecting at point P outside the circle. Additionally, there is another point B on the circumference of the circle.'] then, please answer, In the given diagram, if PA and PB are tangents to circle O at points A and B, and C is a point on circle O, with angle P measuring 40 degrees, what is the measure of angle ACB? | <answer> 70° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point D. The line segment OD connects points O and D.'] then, please answer, In the given diagram, circle O intersects the x-axis at points A and B, and intersects the positive y-axis at point C. If angle DAB measures 25°, what is the degree measure of angle OCD at point C on the circle in the first quadrant? | <answer> 70° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled A, B, and C. Point D is located on the arc AC.'] then, please answer, As shown in the figure, AB is the diameter of circle O. C and D are two points on circle O. If angle ABC = 40°, then the degree of angle BDC is (). | <answer> 50° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to points B and C on the circumference. Point D lies on the extension of line AB, and point E lies on the extension of line AC. Additionally, there is a line segment connecting points B and E.'] then, please answer, In the given diagram, circle O has diameter AB, with perpendicular chord CD. If chord AF is extended and it intersects DC at point E, what is the measure of angle CFE if angle DFC is 48 degrees? | <answer> 66° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, there is another line connecting points D and C.'] then, please answer, As shown in the figure, AB is the diameter of the circle O, and points C and D are on circle O. If angle ABD = 15°, then the size of angle BCD is (). | <answer> 105° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, another line is drawn from point A to point D, which lies on the circumference of the circle.'] then, please answer, If quadrilateral ABCD is inscribed in circle O and AB is the diameter of the circle, with angle AOD measuring 40 degrees and chord AC bisecting angle DAB, what is the measure of angle ADC? | <answer> 125° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point labeled as A. From point A, two tangents are drawn to the circle, intersecting at point B. Point D lies on the circumference of the circle and is connected to points A and B.'] then, please answer, In the given diagram, if AB is the diameter of circle O, AC is a tangent to circle O, and OC intersects circle O at point D with line BD drawn, what is the measure of angle B if angle C is 46 degrees? | <answer> 22° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the line segment AB.'] then, please answer, If quadrilateral ABCD is inscribed in circle O with AB as the diameter, and angle C measures 125°, what is the measure of angle ABD when BD is connected? | <answer> 35° </answer> |
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['The diagram consists of a triangle ABC with vertices A, B, and C. Point D is located on side AB, and point E is located on side AC. Line DE is parallel to side BC.'] then, please answer, In the provided figure, triangle ABC has DE as its median line. If BC has a length of 3.0 units, what is the length of DE? | <answer> 1.5cm </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled as A, B, C, and D. The line segments AB, BC, CD, and DA connect these points in sequence.'] then, please answer, As shown in the diagram, quadrilateral ABCD is circumscribed about circle O. If angle BOD = 164°, the degree measure of angle BCD is (). | <answer> 98° </answer> |
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['The diagram consists of a circle denoted as circle O. On the circumference of the circle, there are three points labeled A, B, and C. Point D is located on the line segment BC. Point E lies outside the circle and is connected to points A and C by line segments AE and CE respectively.'] then, please answer, As shown in the figure, the extended lines of the chords AB and CD of circle O intersect outside the circle at point E. If angle AOC = 100° and angle BCD = 20°, then the size of angle E is (). | <answer> 30° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn: one line intersects the circle at points B and C, while the other line intersects the circle at points D and E. The line segment AB is perpendicular to the line segment AC.'] then, please answer, In the given diagram, quadrilateral ABCD is inscribed in circle O with diagonals AC and BD intersecting at point E, and BD serving as the diameter of circle O. If the measure of angle BDC is 40° and angle AEB is 110°, what is the measure of angle ABC? | <answer> 80° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two tangents are drawn to the circle, intersecting at points B and C. Point D lies on the circumference of the circle. Additionally, there is a line segment connecting points A and O.'] then, please answer, As shown in the diagram, AB is the tangent line of circle O. Point A is the point of tangency. Line OB intersects circle O at point C. Point D is on circle O. Connect AD, CD, OA. If angle B = 20°, then the degree measure of angle ADC is (). | <answer> 35° </answer> |
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['The diagram consists of a circle denoted as circle O. On the circumference of the circle, there are three points labeled A, B, and C.'] then, please answer, As shown in the figure, points A, B, and C are on circle O. angle AOB = 100°, angle OBC = 20°, the degree of angle OAC is () | <answer> 30° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point labeled as A. From point A, two lines are drawn to intersect the circle at points C and D. Point B lies on the extension of line AD.'] then, please answer, In the given diagram, there is a circle O with chords AD and CD. The tangent passing through point C intersects the extension of OA at point B. If angle D measures 29 degrees, what is the measure of angle B? | <answer> 32° </answer> |
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['The diagram consists of a circle labeled as circle O. A line segment AB is tangent to the circle at point A. Point B lies on the circumference of the circle.'] then, please answer, AB is tangent to circle O, and A is the point of tangency. Connect OA and OB. If angle AOB = 70°, then the degree of angle B is () | <answer> 20° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled as A, B, C, and D.'] then, please answer, As shown in the figure, quadrilateral ABCD is inscribed in circle O. AB = CD, A is the midpoint of BD, and angle BDC = 54°. Then, angle ADB is equal to () | <answer> 42° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, there is another line segment connecting points D and B.'] then, please answer, In the given diagram, points A, B, and C lie on circle O. If the measure of angle BAC is 26° and a tangent line to circle O passing through point C intersects the extension of OB at point D, what is the measure of angle D? | <answer> 38° </answer> |
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['The diagram consists of a triangle ABC with vertices A, B, and C. Point D is located on the line segment BC. Point E is positioned on the line segment AD. The diagram also includes several lines and angles within the triangle.'] then, please answer, In the given figure of triangle DABC, point D lies on side BC. It is given that angle DAC measures 30.0 degrees and angle DAB measures 75.0 degrees. Line CE bisects angle ACB and intersects side AB at point E. If we connect points D and E, what is the measure of angle DEC? | <answer> 15° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and C. Point D lies on the extension of line AB.'] then, please answer, If AB is equal to CB in circle O and the parallel line to BC passes through point A, intersecting the tangent of the circle at point D, what is the measure of angle ADC if angle ABC equals 46 degrees? | <answer> 67° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the extended line of AB.'] then, please answer, In the given diagram with circle O, if angle CDB measures 25 degrees and the tangent to circle O passing through point C intersects the extension of AB at point E, what is the degree measure of angle E? | <answer> 40° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled as A, B, C, and D. The line segments AB and CD intersect at point O.'] then, please answer, What is the measure of angle ADC if AB is the diameter of circle O, C and D are points on the circle, and lines AD, CD, AC, and BC are all connected, with angle CAB measuring 35 degrees? | <answer> 55° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. A line segment AB is drawn from point A to point B on the circumference. Another line segment AC is drawn from point A to a point C on the circumference. Point E lies on the line segment AB and divides it into two equal parts.'] then, please answer, As shown in the picture, AB is the diameter of the circle O, and chord CD is perpendicular to AB at point E. If angle OCE = 50°, then angle ABD = () | <answer> 70° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point labeled as A. From point A, two lines are drawn to points B and C on the circumference. Point B lies on the diameter of the circle.'] then, please answer, In the provided figure, if AB is the diameter of circle O and point C lies on circle O, and angle A measures 24°, what is the degree measure of arc BC? | <answer> 48° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, there is a line segment connecting points B and C.'] then, please answer, In the given diagram, with AD as the diameter of the circumcircle circle O of triangle ABC, and angle BAD measuring 58°, what is the measure of angle ACB? | <answer> 32° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn: one line intersects the circle at points B and C, while the other line intersects the circle at points D and M.'] then, please answer, If line segment BM is tangent to circle O at point B and angle MBA is 140°, what is the degree measure of angle ACB? | <answer> 40° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled A, B, and C. Point D is located on the extended line of AB.'] then, please answer, In the figure provided, circle O contains two perpendicular chords, AB and CD. If angle ADC measures 35°, what is the degree measure of angle BAD? | <answer> 55° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, another line is drawn from point A to point D, which lies on the circumference of the circle.'] then, please answer, In the given diagram, circle O has a diameter AB, with points C and D positioned on the circle. If OD is perpendicular to AC and DC is connected, and angle COB measures 30 degrees, what is the measure of angle ACD? | <answer> 37.5° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled as A, B, C, and D.'] then, please answer, What is the measure of angle BDC if points D and A are situated on the minor arc BC and major arc BC of circle O respectively, and angle BOC measures 110°? | <answer> 125° </answer> |
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['The diagram shows a right triangle ABC with the right angle at B. Point D is located on side AC, and point E is located on side AB. The length of side BC is 3 units, and the length of side BD is 2 units.'] then, please answer, In the provided figure, during one night, Xiaoying walked from point B towards point C under the street light A. The length of the shadow CD was measured as 1.0. When she kept walking towards point D, the end E of the shadow DE was measured, and the angle of elevation from Xiaoying to the street light A was found to be 45.0 degrees. It is also known that Xiaoying's height is 1.5 units. What is the height AB of the street light A? | <answer> 4.5米 </answer> |
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['The diagram consists of a circle denoted as circle O. A line segment AB is tangent to the circle at point B. Point C lies on the circumference of the circle, and line segment OC connects points O and C.'] then, please answer, What is the measure of angle CBD if points A, B, and C lie on circle O and point D is located on the extension of line AB, given that angle AOC measures 110 degrees? | <answer> 55° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled as A, B, C, and D. Additionally, there is another point labeled as E on the circumference of the circle.'] then, please answer, In the given illustration, there is an equilateral triangle ABC inscribed within circle O. If triangle ABC is rotated 30° clockwise around the center O, forming triangle DEF, and then BF is connected, what is the measure of angle BFE? | <answer> 15° </answer> |
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['The diagram shows a circle with center O. Point A is on the circumference of the circle, and point B is also on the circumference. Point C is inside the circle. The line segment AB is a chord of the circle.'] then, please answer, If point B lies on AC in circle O and the measure of angle AOC is 100 degrees, what is the degree measure of angle ABC? | <answer> 130° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to points C and D on the circumference. Point B lies on the circumference of the circle.'] then, please answer, As shown in the figure, AB is the diameter of a semicircle, CD is a chord of the semicircle, and CD is parallel to AB. Additionally, angle ACD = 26°. Therefore, angle B is equal to (). | <answer> 64° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Additionally, there is another point labeled as D on the circumference.'] then, please answer, In the given figure, circle O has diameter AB, and points C and D are the trisection points of BE. If angle AOE measures 60°, what is the degree measure of angle BOD? | <answer> 80° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled as A, B, C, and D. Point O is the center of the circle.'] then, please answer, What is the measure of angle CBD if BD is the diameter of circle O, with points A and C lying on circle O and angle A measuring 40 degrees? | <answer> 50° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled A, B, C, and D. Additionally, there is another point labeled E on the circumference of the circle.'] then, please answer, What is the measure of angle BOD if points A, B, C, D, and E lie on circle O, with angle BAC measuring 10 degrees and angle CED measuring 30 degrees? | <answer> 80° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point labeled as A. From point A, two lines are drawn to points B and C on the circumference. Point O is the center of the circle.'] then, please answer, In the given diagram, if points A, B, and C lie on circle O and angle ABC is 50 degrees, what is the measurement of angle AOC? | <answer> 100° </answer> |
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['The diagram consists of a circle denoted as circle O. On the circumference of the circle, there is a point labeled A. From point A, two line segments are drawn to points B and C on the circumference. Point B lies on the circumference of the circle, while point C is not on the circumference but is connected to point A by a line segment.'] then, please answer, What is the measure of angle ACB if angle ABO is 36 degrees and points A, B, and C lie on circle O? | <answer> 54° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled A, B, C, and D. The line segment AB is a diameter of the circle.'] then, please answer, In the given diagram, circle O has AB as its diameter and CD as a chord. If angle ABD measures 54°, what is the measure of angle BCD? | <answer> 36° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the extension of line segment AB.'] then, please answer, In the given illustration, if AB is the diameter of circle O and points C and D lie on circle O, what is the measure of angle D if angle CAB is 36°? | <answer> 54° </answer> |
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['The diagram consists of a circle denoted as circle O. On the circumference of the circle, there is a point labeled A. Additionally, there is another point labeled B on the circumference of the circle.'] then, please answer, If A, B, and C are all points on circle O, and angle AOB measures 60 degrees, what is the measure of angle ACB? | <answer> 30° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the line segment AB.'] then, please answer, If AB is the diameter of circle O and point D is located on circle O, with angle AOC measuring 130°, what is the measure of angle BDC? | <answer> 25° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled as A, B, C, and D.'] then, please answer, As shown in the figure, AB is the diameter of circle O, and angle ACD = 25°. The degree of angle BAD is () | <answer> 65° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled as A, B, C, and D. The line segment AB is a diameter of the circle. Point P lies on the circumference of the circle.'] then, please answer, In the given diagram, chord AB and CD intersect at point P in circle O. If angle A measures 40° and angle B measures 35°, what is the degree measure of angle APD? | <answer> 75° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, another line is drawn from point A to point D on the circumference of the circle.'] then, please answer, In the given diagram, if AB is the diameter of circle O and points C and D lie on circle O, and angle BAC measures 33°, what is the degree measure of angle D? | <answer> 57° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point labeled as A. From point A, two lines are drawn to intersect the circle at points B and C. Additionally, another line is drawn from point A to point D, which lies on the circumference of the circle.'] then, please answer, In the given diagram, circle O has diameter AB and points C and D lie on the circle. If angle CAB measures 50 degrees, what is the degree measure of angle ADC? | <answer> 140° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Additionally, there is another point labeled as E on the circumference of the circle.'] then, please answer, As shown in the figure, AB and EG are the diameters of circle O. EF and EB are chords of circle O, and EF = EB. EF intersects AB at point C. Connect OF. If angle AOF = 20°, then the degree of angle F is (). | <answer> 40° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled A, B, and C. Point D is located on the line segment AB.'] then, please answer, What is the measure of angle A if AB is the diameter of circle O, and points C and D lie on circle O, with OC being parallel to DB and AD and CD are connected? | <answer> 34° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled A, B, and C. Point D is located on the line segment AB.'] then, please answer, In the given diagram, if AB serves as the diameter of circle O and points C and D lie on the circle, with angle DCB measuring 30 degrees, what is the measurement of angle AOD? | <answer> 120° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point O is the center of the circle.'] then, please answer, If angle BAO measures 32 degrees in the given diagram where points A, B, and C lie on circle O, what is the measure of angle ACB? | <answer> 58° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Additionally, there is another point labeled as D on the circumference of the circle. The line segment AB passes through the center of the circle, and the line segment AC intersects the line segment BD at point O.'] then, please answer, As shown in the figure, AB is the diameter of circle O, CD is the chord of circle O. Connect AC, AD, BD. If angle BAC = 34°, then the degree of angle ADC is (). | <answer> 56° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C.'] then, please answer, If angle BOC is measured at 64° in the diagram with points A, B, and C lying on circle O, what is the measure of angle BAC? | <answer> 32° </answer> |
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['The diagram consists of a circle denoted as circle O. On the circumference of the circle, there is a point labeled A. From point A, two lines are drawn to intersect the circle at points B and C.'] then, please answer, What is the measure of angle ACB in circle O if angle OAB is 40 degrees and points A, B, and C lie on the circle? | <answer> 50° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the line segment AB.'] then, please answer, As shown in the figure, points A, B, C are on circle O. The extension of CO intersects AB at point D. angle A = 50° and angle ADC = 110°. The degree measure of angle B is () | <answer> 30° </answer> |
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['The diagram consists of a circle denoted as circle O. On the circumference of the circle, there are three points labeled A, B, and C.'] then, please answer, If the measure of angle AOB is 84 degrees in the given figure where points A, B, and C lie on circle O, what is the degree measure of angle ACB? | <answer> 42° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two lines are drawn to intersect the circle at points B and D. Additionally, there is another line connecting points C and D.'] then, please answer, As shown in the diagram, quadrilateral ABCD is an inscribed quadrilateral of circle O, where BC is the diameter of circle O. Given that angle ACB=18° and D is the midpoint of AC, the measure of angle DAC is (). | <answer> 36° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the extended line of AB.'] then, please answer, As shown in the figure, AB is the diameter of the circle O, and C, D are two points on the circle O. If angle BDC = 52°, then the size of angle ABC is (). | <answer> 38° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point labeled as A. Additionally, there is another point labeled as B on the circumference of the circle.'] then, please answer, In the given illustration, if angle ACB in circle O measures 50° and point P lies on minor arc AB with angle AOP measuring 45°, what is the degree measure of angle PBO? | <answer> 62.5° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there is a point named A. From point A, two tangents are drawn to the circle, intersecting at points B and C on the circumference. Point D lies on the extension of line BC.'] then, please answer, As shown in the diagram, quadrilateral ABCD is an inscribed quadrilateral of circle O, and angle A = 60°. The degree measure of angle DCE is (). | <answer> 60° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled A, B, C, and D. Point E is located on the line segment AB.'] then, please answer, In the given diagram, triangle ABC is inscribed in circle O with angle A measuring 40° and angle ABC measuring 70°. If BD is the diameter of circle O, and it intersects AC at point E, what is the degree measure of angle AEB? | <answer> 120° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are four points labeled as A, B, C, and D in a clockwise direction. The line segment AB is a diameter of the circle.'] then, please answer, In the given illustration, if AB is the diameter of circle O and points C and D lie on the circle, with angle AOC measuring 126°, what is the measure of angle CDB? | <answer> 27° </answer> |
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['The diagram consists of a circle labeled as circle O. On the circumference of the circle, there are three points labeled as A, B, and C. Point D is located on the extension of line segment AB.'] then, please answer, As shown in the figure, AB is the diameter of circle O, C and D are two points on circle O. Given that angle CDB = 20°, the measure of angle ABC is (). | <answer> 70° </answer> |
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