WaltonFuture/GeoQA-8K-in-context-synthesizing

images images listlengths 1 1	problem stringlengths 14 524	answer stringclasses 1 value
	<image>Question: In circle O, BC is the diameter, AB intersects the circle at point E, AC intersects the circle at point D, AD equals CD, and angle B measures 60 degrees. What is the measure of angle BOE?	NA
	<image>Question: If BD is the diameter of circle O and point A and C lie on the circle, with CD being equal to OA, what is the measure of angle BDC?	NA
	<image>In the given diagram, quadrilateral ABCD is inscribed in circle O, and angle C measures 80 degrees. What is the measure of angle AOB?	NA
	<image>If triangle OAD is congruent to triangle OBC and angle O measures 65.0 degrees while angle B measures 30.0 degrees, what is the degree measure of angle OAD?	NA
	<image>If point A, B, and C lie on circle O and angle BAC measures 32 degrees, what is the measure of angle BOC?	NA
	<image>Question: In the given illustration, circle O has diameter AB, with points C and D situated on the circle. If angle ACD measures 42 degrees, what is the degree measurement of angle ADB?	NA
	<image>Question: In the given illustration, if BD is the diameter of circle O and is perpendicular to AC at point E, and angle AOC is 120 degrees, what is the measure of angle BDC?	NA
	<image>Question: In the given illustration, if points A, B, and C lie on circle O and the measure of angle ACB is 59 degrees, what is the degree measure of angle AOB?	NA
	<image>Question: If AB is still the diameter of circle O, and points C and D lie on circle O, but now angle BAC is given as 35 degrees, what is the measure of angle D?	NA
	<image>Question: In circle O, points A, B, C, and D are positioned such that OA is perpendicular to BC at point E. If angle ADC measures 40°, what is the degree measure of angle OBC?	NA
	<image>Question: In the given diagram, AB is the diameter of circle O, and CD is a chord of circle O. If angle BCD measures 74°, what is the measurement of angle ACD?	NA
	<image>Question: In the given diagram, circle O shows radius OC being perpendicular to AB at point H. If angle OAB is 35 degrees, what is the measure of angle ACB?	NA
	<image>Question: In the given circle with center O, AB is the diameter. Points C and D lie on the circumference such that C is the midpoint of arc BAD. Line CD intersects AB at point F. Given that angle ABC measures 30°, determine the measure of angle AOD.	NA
	<image>In the given diagram, triangle ABC is inscribed in circle O with angle A measuring 60°. The midpoint of side BC is denoted by E. What is the measure of angle D formed when extending OE to intersect circle O at point D and connecting BD?	NA
	<image>Given an isosceles triangle ABC with a vertex angle BAC measuring 60°, a semicircle with AB as the diameter intersects BC and AC at points D and E, what is the measure of angle DE?	NA
	<image>Question: In the given diagram, the line segment AB is the diameter of circle O. Chord CD is perpendicular to AB at point E. If AC, OC, and OD are connected, and angle COD measures 60 degrees, what is the measure of angle A?	NA
	<image>Question: In circle O, AB is a diameter, and C and D are two points on circle O. OC is parallel to AD. If angle BAC = 30°, then the degree of angle BOC is ().	NA
	<image>Question: If AB is the diameter of a semicircle and C and D are points on the semicircle such that angle CAD measures 30 degrees, what is the measure of angle BDC?	NA
	<image>Question: In circle O, AB is the diameter, angle A = 60°, point D is the midpoint of chord AC, and point E is any point on BC. What is the range of possible values for angle CED?	NA
	<image>In the given diagram, the line AB is a tangent to circle O at point A, while OB intersects circle O at point C, and point D lies on the major arc AC. If angle B measures 45 degrees, what is the degree measure of angle ADC?	NA
	<image>In the given diagram, quadrilateral ABCD is inscribed in circle O. If the measure of angle B is 110 degrees, what is the degree measure of angle D?	NA
	<image>Question: In the given diagram, the chords AB and CD intersect at point E, and the measure of arc AD is equal to the measure of arc CB. If angle A measures 60 degrees, what is the degree measure of angle CEB?	NA
	<image>In the figure, triangle ABC is circumscribed about circle O, AB = AC, BO is connected and extended to intersect BC at point D. If angle A = 60°, then the measure of angle BDC is ( ).	NA
	<image>Question: If points A, B, C, and D lie on circle O, with AB equal to AC and BD being a diameter of circle O, and CD being connected, what is the measure of angle BCD if angle A measures 36°?	NA
	<image>Question: In the given circle O, if angle AOB measures 120 degrees, what is the measure of angle BCD?	NA
	<image>Question: In the given diagram, if AB is a chord of circle O and OC is perpendicular to AB, intersecting circle O at point C, and angle AOB measures 120 degrees, what is the degree measure of angle ACB?	NA
	<image>Question: In the given diagram, the line AB is tangent to circle O at point A, and the line OB intersects circle O at point C. Additionally, point D lies on circle O. If angle BAC is measured at 30 degrees, what is the degree measure of angle ADC?	NA
	<image>Question: If point P lies on line segment ED and circle O is the circumcircle of the regular pentagon ABCDE, what is the measure of angle APE?	NA
	<image>In the provided diagram, circle O has diameter AB and point C lies on the circle. If angle ACO measures 30.0 degrees, what is the measure of angle BOC?	NA
	<image>Question: If AB is the diameter of circle O and points C and D lie on circle O, with OD being perpendicular to AC, and angle AOB measures 140°, what is the degree measure of angle BCD?	NA
	<image>Question: If PA and PB are tangents to circle O at points A and B, and C is a moving point on arc AB, what is the measure of angle ACB if angle PAB measures 50 degrees?	NA
	<image>Question: Points A, B, C, and D lie on circle O, with O as the center. The quadrilateral OABC is formed by connecting AD, BD, and consecutively connecting O, A, B, C. Given that OB is equal to AC, and OD is equal to AB, what is the measure of angle D?	NA
	<image>In the given diagram, the line segment AB represents the diameter of circle O. A tangent, DA, is drawn from point D to circle O at point A, and the line DO intersects circle O at point C. If angle D measures 40 degrees, what is the measure of angle ACB?	NA
	<image>Question: In the given illustration, circle O has diameter AB. Points C and D are situated on circle O, on opposite sides of AB. Segments OC, CD, and DA are connected. If angle AOB measures 110 degrees, what is the measure of angle C?	NA
	<image>If quadrilateral ADCB is inscribed in circle O and angle ADB measures 58 degrees, what is the measure of angle ACB?	NA
	<image>Question: In the given diagram, AB is the diameter of circle O. CB is tangent to circle O at point B. AC intersects circle O at point D. OD is connected. If angle ADB = 72°, find the measure of angle BOD.	NA
	<image>Question: If PA is a tangent to circle O at point A and the extension of PO intersects circle O at point C, and if angle P measures 30 degrees, what is the measure of angle CQA when point Q is a moving point on line AC?	NA
	<image>Question: In triangle ABC inscribed in circle O, the bisector of the external angle BAE intersects circle O at point D, and A is the midpoint of arc CD. If angle ABC measures 30°, what is the measure of angle ACB?	NA
	<image>Question: In circle O, point Q lies outside the circle. QA and QB are tangents to circle O at points A and B, respectively. Points C and D are also on circle O. If angle Q measures 130°, what is the sum of angles B and D?	NA
	<image>Question: In the given diagram, PA and PB are tangents to circle O at points A and B respectively. If angle P measures 108 degrees, what is the measure of angle ACB?	NA
	<image>In the given illustration of circle O, where AB, BC, and CD are all equal, and angle AOB measures 60 degrees, what is the degree measure of angle CAD?	NA
	<image>If AB is the diameter of circle O and points C and D lie on the circle, with angle BAC measuring 40 degrees, what is the measure of angle ADC?	NA
	<image>Question: In the given diagram, the line AB is tangent to circle O at point A. Line OB intersects circle O at point C, and point D lies on circle O. If angle BAC measures 45 degrees, what is the measure of angle ABO?	NA
	<image>In the given illustration, in circle O, the radius AO is perpendicular to OB. Point P lies on the major arc APB, and the midpoint of AB is denoted as C. If we connect AP and CP, what is the measure of angle ACP?	NA
	<image>Question: In the given diagram, four points A, B, C, and D lie on circle O. The measure of angle AOC is 120 degrees and point B is the midpoint of AC. What is the degree measure of angle D?	NA
	<image>Question: If AB is the diameter of circle O, point C lies on circle O with CO perpendicular to AB at point O, and chord CD intersects AB at point E with angle AEC measuring 64°, what is the measure of angle BCD?	NA
	<image>Question: If triangle ABC is inscribed in circle O, angle BAC measures 60 degrees, and AD is drawn parallel to BC passing through point A, intersecting the extension of BO at point D, what is the measure of angle D?	NA
	<image>Question: If points A, B, C, and D lie on circle O, and lines AB and CD intersect at an external point E outside circle O, with angle BCD measuring 30° and angle E measuring 45°, what is the measure of angle ADC?	NA
	<image>Question: If triangle ABC undergoes a counterclockwise rotation around point A by 56.0 degrees to form triangle ADE, and angle BAE is measured at 112.0 degrees, what is the degree measure of angle DAE?	NA
	<image>In the given illustration, within circle O, point D serves as the midpoint of line segment AB, where CD represents the diameter of circle O. Additionally, line segment AE is parallel to BC and intersects circle O at point E. If angle AEB measures 60 degrees, what is the measure of angle DCB?	NA
	<image>Question: In the given diagram, BD is the diameter of circle O. Points A and C are on circle O, with AB = AD. AC intersects BD at point G. If angle BOD = 140°, find the degree measure of angle AGB.	NA
	<image>Question: In the given diagram, the tangent PA touches circle O at point A. If the line PO intersects circle O at point B, and point C lies on circle O, and angle P measures 60 degrees, what is the degree measure of angle ACB?	NA
	<image>Question: In the same diagram showing triangle ABC inscribed in circle O, if angle BAC measures 45 degrees, what is the measure of angle BOC?	NA
	<image>Question: If angle ABO measures 30 degrees and O is the circumcenter of triangle ABC, what is the measure of angle ACB?	NA
	<image>Question: If angle BAC measures 45 degrees and point O lies on side AB, with circle O being tangent to side AC at point D and intersecting side AB at points E and F, what is the measure of angle AFD?	NA
	<image>Question: In the given diagram, triangle ABD has its circumcircle denoted as circle O, with AC serving as the diameter. If the measure of angle ADB is 60°, what is the measure of angle ABC?	NA
	<image>Question: In the given diagram, triangle ABC is inscribed in circle O, and AD is the diameter of the circle. If angle ABD measures 60 degrees, what is the degree measure of angle ACB?	NA
	<image>Question: In the given figure, triangle BCD is inscribed in circle O, and angle D measures 70 degrees. Line OA is perpendicular to BC and intersects circle O at point A. If angle BAC is bisected by line AD, what is the measure of angle BAD?	NA
	<image>Question: In the given illustration, circle O has diameter AB, and points C and D lie on the circle. If angle BCD measures 72°, what is the degree measure of angle A?	NA
	<image>Question: If the line segment OA is rotated clockwise around point O by 60 degrees, what is the new angle measure of BOA' given that angle B measures 50 degrees and BA is tangent to circle O at point A?	NA
	<image>In the given diagram, triangle ABC is inscribed in circle O. If the measure of angle OCA is 40°, what is the measure of angle BAC?	NA
	<image>In the given diagram, circle O is the circumcircle of triangle ABC. If angle OBA measures 30°, what is the measure of angle ACB?	NA
	<image>Question: In the given figure, triangle ABC is inscribed in circle O with a radius of 3. If angle ACB measures 60°, what is the length of side AB?	NA
	<image>Question: In the given diagram, triangle ABC is inscribed in circle O, with the measure of angle BAC being 74 degrees. If point D lies on the arc BC such that BD = DC, what is the measure of angle BDC?	NA
	<image>Question: If chord CD intersects diameter AB at point E in circle O, and angle ACD is 30 degrees while angle BDC is 50 degrees, what is the measure of angle COB?	NA
	<image>If points A, B, and C are all located on circle O and the measure of angle OAB is 40 degrees, what is the measurement of angle ACB?	NA
	<image>Question: If AB is the diameter of circle O and point C lies on circle O, what can be inferred about the relationship between angle ACB and angle AOB?	NA
	<image>In the given figure, circle O is intersected by OB at point C, with AB being a tangent to the circle at point A. Additionally, point D lies on circle O and lines AD, CD, and OA are connected. If angle BAC measures 30°, what is the degree measure of angle ABO?	NA
	<image>In the given diagram, quadrilateral ABCD is inscribed in circle O, and one of its interior angles, angle ADC, measures 74°. What is the measure of angle AOC?	NA
	<image>In the given diagram, circle O has diameter AB, and points C and D lie on the circle. If angle ABD measures 75° and angle DCO measures 18°, what is the degree measure of angle BDC?	NA
	<image>Question: If point C is the midpoint of chord AB in circle O, and angle COB measures 70 degrees, what is the measure of angle ADB?	NA
	<image>Question: If points C and D are located on a semicircle with AB as the diameter and angle ADC measures 150 degrees, and point E is an arbitrary point on AD, what is the measure of angle BEC when BE and CE are connected?	NA
	<image>If angle BAC is 30 degrees and points A, B, and C are all located on circle O, what is the measure of angle BOC?	NA
	<image>If points A, B, and C lie on circle O, and angle ACB measures 57 degrees, what is the measure of angle AOB?	NA
	<image>Question: If BC is the diameter of circle O and AB is a chord of circle O, what is the relationship between the measure of angle ABC and the measure of angle AOC when angle AOC measures 90 degrees?	NA
	<image>Question: If AB is the diameter of circle O, D is the midpoint of the semicircle, and C is any point on the other half of the circle, what geometric relationships can be observed between the segments OD, CD, and BC?	NA
	<image>Question: In the given figure, a tangent line AB touches circle O at point A, and line BO intersects circle O at point C. Additionally, point D is on circle O. If the measure of angle BAC is 27°, what is the measure of angle B?	NA
	<image>Question: If triangle ABC is circumscribed about circle O, angle A measures 60 degrees, and OD is perpendicular to BC with the foot at E, what is the measure of angle CBD?	NA
	<image>In the given diagram, all points A, B, and C lie on circle O. If the measure of angle ACB is 70 degrees, what is the measure of angle AOB?	NA
	<image>Question: If AB is the diameter of circle O and points C, D, and E lie on circle O, with angle ABE measuring 30°, what is the degree measure of angle BDE?	NA
	<image>In the figure, if the circle O is tangent to the two sides AD and BE of the regular pentagon ABCDE at points D and E, what is the measure of angle AOE?	NA
	<image>Question: In the given illustration, the quadrilateral ABCD is circumscribed in the circle O. A point P, which is not coincident with points A and D, is arbitrarily chosen on side AD. If angle B is 120 degrees and angle D is 80 degrees, what could be the measure of angle APC?	NA
	<image>In the provided figure, circle O has diameter AB and tangent BC. If angle ABC measures 40°, what is the measure of angle ACB?	NA
	<image>Question: In the given diagram, quadrilateral ABCD is inscribed in circle O. The diameter BE is part of circle O and AE is connected to it. If the measure of angle ABE is 60°, and the measure of angle BCD is twice the measure of angle BAD, then what is the degree measure of angle DAE?	NA
	<image>Question: In the given diagram, line AB is tangent to circle O at point C, and line AO meets circle O at point D. If angle AOC measures 60°, what is the degree measurement of angle BCD?	NA
	<image>In the given illustration, lines PA and PB are tangents to circle O at points A and B. If angle P measures 80°, what is the measure of angle ABO?	NA
	<image>Question: In circle O, AB is the tangent at point A, OB intersects circle O at point C, and a line parallel to OB passing through point A intersects circle O at point D. If angle B measures 60 degrees, what is the measure of angle OCD?	NA
	<image>In the given figure, if AB is parallel to CD and AD bisects angle BAE, with angle D measuring 42.0 degrees, what is the degree measure of angle AEC?	NA
	<image>Question: If points A and D are on diameter BC of circle O, and angle ADB measures 30 degrees, what is the measure of angle ACB?	NA
	<image>In the given diagram, points A, B, and C lie on the circumference of circle O with a radius of 1. If the length of AB is √2 and the measure of angle ACB is 45 degrees, what is the degree measure of angle ABC?	NA
	<image>Question: If AB and CD are perpendicular chords of circle O, intersecting at point P, and angle ABC measures 60 degrees, with point E lying on the circle, what is the measure of angle CED?	NA
	<image>In the given diagram, circle O has diameter AB and point C lies on the circle. If angle BAC measures 34 degrees, what is the measure of angle BOC?	NA
	<image>Question: If AB is the diameter of circle O, point C lies on circle O, and angle AOB measures 120 degrees, what is the measure of angle ACB?	NA
	<image>Question: If points A, B, and C are on circle O and angle BAC measures 72°, what is the measure of angle BOC?	NA
	<image>In the given diagram, quadrilateral ABCD is inscribed in circle O. If the interior angle A measures 50°, what is the degree measurement of angle BOD?	NA
	<image>Question: In the given circle with center O, points A, B, and C lie on the circumference. If angle AOC measures 120° and angle BAC measures 45°, what is the measure of angle AOB?	NA
	<image>If line segment AB serves as the diameter of circle O, and chord CD is perpendicular to AB at point E, such that the length of CD is three times the length of OE, what is the measure of angle BCD?	NA
	<image>Question: If the lines PA and PB touch circle O at points A and B, respectively, and angle ACB measures 125 degrees, what is the measure of angle P?	NA
	<image>Question: Given points A, B, C, D, and E on circle O, where AB is equal to CD and angle AOB measures 60 degrees, what is the measure of angle CED?	NA
	<image>Question: In the given diagram, triangle ABC is inscribed in circle O with AD as the diameter. If angle BAC measures 60 degrees and OA equals 3, find the length of the minor arc CD.	NA

<image>Question: In circle O, BC is the diameter, AB intersects the circle at point E, AC intersects the circle at point D, AD equals CD, and angle B measures 60 degrees. What is the measure of angle BOE?

NA

<image>Question: If BD is the diameter of circle O and point A and C lie on the circle, with CD being equal to OA, what is the measure of angle BDC?

NA

<image>In the given diagram, quadrilateral ABCD is inscribed in circle O, and angle C measures 80 degrees. What is the measure of angle AOB?

NA

<image>If triangle OAD is congruent to triangle OBC and angle O measures 65.0 degrees while angle B measures 30.0 degrees, what is the degree measure of angle OAD?

NA

<image>If point A, B, and C lie on circle O and angle BAC measures 32 degrees, what is the measure of angle BOC?

NA

<image>Question: In the given illustration, circle O has diameter AB, with points C and D situated on the circle. If angle ACD measures 42 degrees, what is the degree measurement of angle ADB?

NA

<image>Question: In the given illustration, if BD is the diameter of circle O and is perpendicular to AC at point E, and angle AOC is 120 degrees, what is the measure of angle BDC?

NA

<image>Question: In the given illustration, if points A, B, and C lie on circle O and the measure of angle ACB is 59 degrees, what is the degree measure of angle AOB?

NA

<image>Question: If AB is still the diameter of circle O, and points C and D lie on circle O, but now angle BAC is given as 35 degrees, what is the measure of angle D?

NA

<image>Question: In circle O, points A, B, C, and D are positioned such that OA is perpendicular to BC at point E. If angle ADC measures 40°, what is the degree measure of angle OBC?

NA

<image>Question: In the given diagram, AB is the diameter of circle O, and CD is a chord of circle O. If angle BCD measures 74°, what is the measurement of angle ACD?

NA

<image>Question: In the given diagram, circle O shows radius OC being perpendicular to AB at point H. If angle OAB is 35 degrees, what is the measure of angle ACB?

NA

<image>Question: In the given circle with center O, AB is the diameter. Points C and D lie on the circumference such that C is the midpoint of arc BAD. Line CD intersects AB at point F. Given that angle ABC measures 30°, determine the measure of angle AOD.

NA

<image>In the given diagram, triangle ABC is inscribed in circle O with angle A measuring 60°. The midpoint of side BC is denoted by E. What is the measure of angle D formed when extending OE to intersect circle O at point D and connecting BD?

NA

<image>Given an isosceles triangle ABC with a vertex angle BAC measuring 60°, a semicircle with AB as the diameter intersects BC and AC at points D and E, what is the measure of angle DE?

NA

<image>Question: In the given diagram, the line segment AB is the diameter of circle O. Chord CD is perpendicular to AB at point E. If AC, OC, and OD are connected, and angle COD measures 60 degrees, what is the measure of angle A?

NA

<image>Question: In circle O, AB is a diameter, and C and D are two points on circle O. OC is parallel to AD. If angle BAC = 30°, then the degree of angle BOC is ().

NA

<image>Question: If AB is the diameter of a semicircle and C and D are points on the semicircle such that angle CAD measures 30 degrees, what is the measure of angle BDC?

NA

<image>Question: In circle O, AB is the diameter, angle A = 60°, point D is the midpoint of chord AC, and point E is any point on BC. What is the range of possible values for angle CED?

NA

<image>In the given diagram, the line AB is a tangent to circle O at point A, while OB intersects circle O at point C, and point D lies on the major arc AC. If angle B measures 45 degrees, what is the degree measure of angle ADC?

NA

<image>In the given diagram, quadrilateral ABCD is inscribed in circle O. If the measure of angle B is 110 degrees, what is the degree measure of angle D?

NA

<image>Question: In the given diagram, the chords AB and CD intersect at point E, and the measure of arc AD is equal to the measure of arc CB. If angle A measures 60 degrees, what is the degree measure of angle CEB?

NA

<image>In the figure, triangle ABC is circumscribed about circle O, AB = AC, BO is connected and extended to intersect BC at point D. If angle A = 60°, then the measure of angle BDC is ( ).

NA

<image>Question: If points A, B, C, and D lie on circle O, with AB equal to AC and BD being a diameter of circle O, and CD being connected, what is the measure of angle BCD if angle A measures 36°?

NA

<image>Question: In the given circle O, if angle AOB measures 120 degrees, what is the measure of angle BCD?

NA

<image>Question: In the given diagram, if AB is a chord of circle O and OC is perpendicular to AB, intersecting circle O at point C, and angle AOB measures 120 degrees, what is the degree measure of angle ACB?

NA

<image>Question: In the given diagram, the line AB is tangent to circle O at point A, and the line OB intersects circle O at point C. Additionally, point D lies on circle O. If angle BAC is measured at 30 degrees, what is the degree measure of angle ADC?

NA

<image>Question: If point P lies on line segment ED and circle O is the circumcircle of the regular pentagon ABCDE, what is the measure of angle APE?

NA

<image>In the provided diagram, circle O has diameter AB and point C lies on the circle. If angle ACO measures 30.0 degrees, what is the measure of angle BOC?

NA

<image>Question: If AB is the diameter of circle O and points C and D lie on circle O, with OD being perpendicular to AC, and angle AOB measures 140°, what is the degree measure of angle BCD?

NA

<image>Question: If PA and PB are tangents to circle O at points A and B, and C is a moving point on arc AB, what is the measure of angle ACB if angle PAB measures 50 degrees?

NA

<image>Question: Points A, B, C, and D lie on circle O, with O as the center. The quadrilateral OABC is formed by connecting AD, BD, and consecutively connecting O, A, B, C. Given that OB is equal to AC, and OD is equal to AB, what is the measure of angle D?

NA

<image>In the given diagram, the line segment AB represents the diameter of circle O. A tangent, DA, is drawn from point D to circle O at point A, and the line DO intersects circle O at point C. If angle D measures 40 degrees, what is the measure of angle ACB?

NA

<image>Question: In the given illustration, circle O has diameter AB. Points C and D are situated on circle O, on opposite sides of AB. Segments OC, CD, and DA are connected. If angle AOB measures 110 degrees, what is the measure of angle C?

NA

<image>If quadrilateral ADCB is inscribed in circle O and angle ADB measures 58 degrees, what is the measure of angle ACB?

NA

<image>Question: In the given diagram, AB is the diameter of circle O. CB is tangent to circle O at point B. AC intersects circle O at point D. OD is connected. If angle ADB = 72°, find the measure of angle BOD.

NA

<image>Question: If PA is a tangent to circle O at point A and the extension of PO intersects circle O at point C, and if angle P measures 30 degrees, what is the measure of angle CQA when point Q is a moving point on line AC?

NA

<image>Question: In triangle ABC inscribed in circle O, the bisector of the external angle BAE intersects circle O at point D, and A is the midpoint of arc CD. If angle ABC measures 30°, what is the measure of angle ACB?

NA

<image>Question: In circle O, point Q lies outside the circle. QA and QB are tangents to circle O at points A and B, respectively. Points C and D are also on circle O. If angle Q measures 130°, what is the sum of angles B and D?

NA

<image>Question: In the given diagram, PA and PB are tangents to circle O at points A and B respectively. If angle P measures 108 degrees, what is the measure of angle ACB?

NA

<image>In the given illustration of circle O, where AB, BC, and CD are all equal, and angle AOB measures 60 degrees, what is the degree measure of angle CAD?

NA

<image>If AB is the diameter of circle O and points C and D lie on the circle, with angle BAC measuring 40 degrees, what is the measure of angle ADC?

NA

<image>Question: In the given diagram, the line AB is tangent to circle O at point A. Line OB intersects circle O at point C, and point D lies on circle O. If angle BAC measures 45 degrees, what is the measure of angle ABO?

NA

<image>In the given illustration, in circle O, the radius AO is perpendicular to OB. Point P lies on the major arc APB, and the midpoint of AB is denoted as C. If we connect AP and CP, what is the measure of angle ACP?

NA

<image>Question: In the given diagram, four points A, B, C, and D lie on circle O. The measure of angle AOC is 120 degrees and point B is the midpoint of AC. What is the degree measure of angle D?

NA

<image>Question: If AB is the diameter of circle O, point C lies on circle O with CO perpendicular to AB at point O, and chord CD intersects AB at point E with angle AEC measuring 64°, what is the measure of angle BCD?

NA

<image>Question: If triangle ABC is inscribed in circle O, angle BAC measures 60 degrees, and AD is drawn parallel to BC passing through point A, intersecting the extension of BO at point D, what is the measure of angle D?

NA

<image>Question: If points A, B, C, and D lie on circle O, and lines AB and CD intersect at an external point E outside circle O, with angle BCD measuring 30° and angle E measuring 45°, what is the measure of angle ADC?

NA

<image>Question: If triangle ABC undergoes a counterclockwise rotation around point A by 56.0 degrees to form triangle ADE, and angle BAE is measured at 112.0 degrees, what is the degree measure of angle DAE?

NA

<image>In the given illustration, within circle O, point D serves as the midpoint of line segment AB, where CD represents the diameter of circle O. Additionally, line segment AE is parallel to BC and intersects circle O at point E. If angle AEB measures 60 degrees, what is the measure of angle DCB?

NA

<image>Question: In the given diagram, BD is the diameter of circle O. Points A and C are on circle O, with AB = AD. AC intersects BD at point G. If angle BOD = 140°, find the degree measure of angle AGB.

NA

<image>Question: In the given diagram, the tangent PA touches circle O at point A. If the line PO intersects circle O at point B, and point C lies on circle O, and angle P measures 60 degrees, what is the degree measure of angle ACB?

NA

<image>Question: In the same diagram showing triangle ABC inscribed in circle O, if angle BAC measures 45 degrees, what is the measure of angle BOC?

NA

<image>Question: If angle ABO measures 30 degrees and O is the circumcenter of triangle ABC, what is the measure of angle ACB?

NA

<image>Question: If angle BAC measures 45 degrees and point O lies on side AB, with circle O being tangent to side AC at point D and intersecting side AB at points E and F, what is the measure of angle AFD?

NA

<image>Question: In the given diagram, triangle ABD has its circumcircle denoted as circle O, with AC serving as the diameter. If the measure of angle ADB is 60°, what is the measure of angle ABC?

NA

<image>Question: In the given diagram, triangle ABC is inscribed in circle O, and AD is the diameter of the circle. If angle ABD measures 60 degrees, what is the degree measure of angle ACB?

NA

<image>Question: In the given figure, triangle BCD is inscribed in circle O, and angle D measures 70 degrees. Line OA is perpendicular to BC and intersects circle O at point A. If angle BAC is bisected by line AD, what is the measure of angle BAD?

NA

<image>Question: In the given illustration, circle O has diameter AB, and points C and D lie on the circle. If angle BCD measures 72°, what is the degree measure of angle A?

NA

<image>Question: If the line segment OA is rotated clockwise around point O by 60 degrees, what is the new angle measure of BOA' given that angle B measures 50 degrees and BA is tangent to circle O at point A?

NA

<image>In the given diagram, triangle ABC is inscribed in circle O. If the measure of angle OCA is 40°, what is the measure of angle BAC?

NA

<image>In the given diagram, circle O is the circumcircle of triangle ABC. If angle OBA measures 30°, what is the measure of angle ACB?

NA

<image>Question: In the given figure, triangle ABC is inscribed in circle O with a radius of 3. If angle ACB measures 60°, what is the length of side AB?

NA

<image>Question: In the given diagram, triangle ABC is inscribed in circle O, with the measure of angle BAC being 74 degrees. If point D lies on the arc BC such that BD = DC, what is the measure of angle BDC?

NA

<image>Question: If chord CD intersects diameter AB at point E in circle O, and angle ACD is 30 degrees while angle BDC is 50 degrees, what is the measure of angle COB?

NA

<image>If points A, B, and C are all located on circle O and the measure of angle OAB is 40 degrees, what is the measurement of angle ACB?

NA

<image>Question: If AB is the diameter of circle O and point C lies on circle O, what can be inferred about the relationship between angle ACB and angle AOB?

NA

<image>In the given figure, circle O is intersected by OB at point C, with AB being a tangent to the circle at point A. Additionally, point D lies on circle O and lines AD, CD, and OA are connected. If angle BAC measures 30°, what is the degree measure of angle ABO?

NA

<image>In the given diagram, quadrilateral ABCD is inscribed in circle O, and one of its interior angles, angle ADC, measures 74°. What is the measure of angle AOC?

NA

<image>In the given diagram, circle O has diameter AB, and points C and D lie on the circle. If angle ABD measures 75° and angle DCO measures 18°, what is the degree measure of angle BDC?

NA

<image>Question: If point C is the midpoint of chord AB in circle O, and angle COB measures 70 degrees, what is the measure of angle ADB?

NA

<image>Question: If points C and D are located on a semicircle with AB as the diameter and angle ADC measures 150 degrees, and point E is an arbitrary point on AD, what is the measure of angle BEC when BE and CE are connected?

NA

<image>If angle BAC is 30 degrees and points A, B, and C are all located on circle O, what is the measure of angle BOC?

NA

<image>If points A, B, and C lie on circle O, and angle ACB measures 57 degrees, what is the measure of angle AOB?

NA

<image>Question: If BC is the diameter of circle O and AB is a chord of circle O, what is the relationship between the measure of angle ABC and the measure of angle AOC when angle AOC measures 90 degrees?

NA

<image>Question: If AB is the diameter of circle O, D is the midpoint of the semicircle, and C is any point on the other half of the circle, what geometric relationships can be observed between the segments OD, CD, and BC?

NA

<image>Question: In the given figure, a tangent line AB touches circle O at point A, and line BO intersects circle O at point C. Additionally, point D is on circle O. If the measure of angle BAC is 27°, what is the measure of angle B?

NA

<image>Question: If triangle ABC is circumscribed about circle O, angle A measures 60 degrees, and OD is perpendicular to BC with the foot at E, what is the measure of angle CBD?

NA

<image>In the given diagram, all points A, B, and C lie on circle O. If the measure of angle ACB is 70 degrees, what is the measure of angle AOB?

NA

<image>Question: If AB is the diameter of circle O and points C, D, and E lie on circle O, with angle ABE measuring 30°, what is the degree measure of angle BDE?

NA

<image>In the figure, if the circle O is tangent to the two sides AD and BE of the regular pentagon ABCDE at points D and E, what is the measure of angle AOE?

NA

<image>Question: In the given illustration, the quadrilateral ABCD is circumscribed in the circle O. A point P, which is not coincident with points A and D, is arbitrarily chosen on side AD. If angle B is 120 degrees and angle D is 80 degrees, what could be the measure of angle APC?

NA

<image>In the provided figure, circle O has diameter AB and tangent BC. If angle ABC measures 40°, what is the measure of angle ACB?

NA

<image>Question: In the given diagram, quadrilateral ABCD is inscribed in circle O. The diameter BE is part of circle O and AE is connected to it. If the measure of angle ABE is 60°, and the measure of angle BCD is twice the measure of angle BAD, then what is the degree measure of angle DAE?

NA

<image>Question: In the given diagram, line AB is tangent to circle O at point C, and line AO meets circle O at point D. If angle AOC measures 60°, what is the degree measurement of angle BCD?

NA

<image>In the given illustration, lines PA and PB are tangents to circle O at points A and B. If angle P measures 80°, what is the measure of angle ABO?

NA

<image>Question: In circle O, AB is the tangent at point A, OB intersects circle O at point C, and a line parallel to OB passing through point A intersects circle O at point D. If angle B measures 60 degrees, what is the measure of angle OCD?

NA

<image>In the given figure, if AB is parallel to CD and AD bisects angle BAE, with angle D measuring 42.0 degrees, what is the degree measure of angle AEC?

NA

<image>Question: If points A and D are on diameter BC of circle O, and angle ADB measures 30 degrees, what is the measure of angle ACB?

NA

<image>In the given diagram, points A, B, and C lie on the circumference of circle O with a radius of 1. If the length of AB is √2 and the measure of angle ACB is 45 degrees, what is the degree measure of angle ABC?

NA

<image>Question: If AB and CD are perpendicular chords of circle O, intersecting at point P, and angle ABC measures 60 degrees, with point E lying on the circle, what is the measure of angle CED?

NA

<image>In the given diagram, circle O has diameter AB and point C lies on the circle. If angle BAC measures 34 degrees, what is the measure of angle BOC?

NA

<image>Question: If AB is the diameter of circle O, point C lies on circle O, and angle AOB measures 120 degrees, what is the measure of angle ACB?

NA

<image>Question: If points A, B, and C are on circle O and angle BAC measures 72°, what is the measure of angle BOC?

NA

<image>In the given diagram, quadrilateral ABCD is inscribed in circle O. If the interior angle A measures 50°, what is the degree measurement of angle BOD?

NA

<image>Question: In the given circle with center O, points A, B, and C lie on the circumference. If angle AOC measures 120° and angle BAC measures 45°, what is the measure of angle AOB?

NA

<image>If line segment AB serves as the diameter of circle O, and chord CD is perpendicular to AB at point E, such that the length of CD is three times the length of OE, what is the measure of angle BCD?

NA

<image>Question: If the lines PA and PB touch circle O at points A and B, respectively, and angle ACB measures 125 degrees, what is the measure of angle P?

NA

<image>Question: Given points A, B, C, D, and E on circle O, where AB is equal to CD and angle AOB measures 60 degrees, what is the measure of angle CED?

NA

<image>Question: In the given diagram, triangle ABC is inscribed in circle O with AD as the diameter. If angle BAC measures 60 degrees and OA equals 3, find the length of the minor arc CD.

NA