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

images images listlengths 1 1	problem stringlengths 14 524	answer stringclasses 1 value
	<image>Question: In the given diagram, square ABCD is enclosed within circle O, and point P is located on the side AD. What is the measure of angle BPC?	NA
	<image>Question: In the given figure, AB is the diameter of the semicircle O. Point C lies on the semicircle O, distinct from points A and B. D is the midpoint of AC. Extend DC to intersect the extension line of AB at point E. If angle CAE measures 20°, what is the measure of angle E?	NA
	<image>In the given diagram of circle O, where AB is the diameter and CD is a chord, if angle BCD measures 60°, what is the degree measure of angle CAD?	NA
	<image>Question: In the given circle with center O, points A, B, and C lie on the circumference. If angle AOB measures 70° and angle BOC measures 40°, what is the measure of angle ACB?	NA
	<image>Question: Given that AB is the diameter of circle O and points C and D lie on circle O, with angle ACD measuring 30 degrees, what is the measure of angle DOB?	NA
	<image>Question: In the same diagram, AB is the diameter of circle O, and C, D are two points on the circle. Given that angle ADB = 60°, find the measure of angle BOC.	NA
	<image>Question: In the given illustration, circle O has diameter AB, and chords AC and BC. If angle C measures 40°, what is the measure of angle A?	NA
	<image>Question: In the given diagram, points A, B, and C lie on circle O. If the measure of angle AOB is 120 degrees, what is the measure of angle ACB?	NA
	<image>If point O is the incenter of triangle ABC and angle A measures 50 degrees, what is the measure of angle BOC when BO and CO are connected?	NA
	<image>If quadrilateral ABCD is inscribed in circle O and angle B measures 60 degrees, what is the degree measure of angle D?	NA
	<image>Question: In the given diagram, triangle ABC is inscribed in circle O, with CD as the diameter of circle O. If angle BAC = 30°, find the measure of angle DBC.	NA
	<image>Question: In the given figure, circle O is the circumcircle of equilateral triangle ABC. Point D is a point on arc BC, and angle BAC = 30°. What is the measure of angle ACD?	NA
	<image>Question: If circle O is the circumcircle of triangle ABC, angle A measures 60 degrees, and E is the midpoint of side BC, what is the measure of angle BCD when OE is extended to intersect circle O at point D and BD is connected?	NA
	<image>Question: In the given diagram, circle O has diameter AB, and points C and D lie on circle O. If angle ADB measures 52°, what is the degree measure of angle ACB?	NA
	<image>Question: 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 AOB if angle B measures 52 degrees?	NA
	<image>Question: If points A, B, and C lie on circle O and angle AOB measures 128°, what is the degree measure of angle ACB?	NA
	<image>Question: 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 ADE if angle BAD is 45 degrees?	NA
	<image>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 ACD measures 25°, what is the degree measure of angle ADC?	NA
	<image>Question: If AB is the diameter of circle O, 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, what is the measure of angle ACD?	NA
	<image>Question: In the given diagram, AB is the diameter of circle O. EF and EB are chords of circle O. If angle AOF measures 50°, what is the measure of angle E?	NA
	<image>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 BAC is 60 degrees, what is the measure of angle OPC?	NA
	<image>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 BDC?	NA
	<image>Question: 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 BCD measuring 40°, what is the measure of angle BAC?	NA
	<image>Question: If point Q lies on the circumcircle of regular hexagon ABCDEF, which is circle O, and Q is not coincident with A or B, what is the measure of angle AQD?	NA
	<image>Question: 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 124 degrees, what is the measure of angle P?	NA
	<image>Question: 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 45°, what is the measure of angle D?	NA
	<image>Question: If BC is the diameter of circle O, and points A and D lie on circle O, with angle BAC measuring 40°, what is the measure of angle ADC?	NA
	<image>Question: In the given circle with center O, AB is the diameter. Point C lies on the circumference, and D is the midpoint of arc AC. If angle AOC measures 120°, what is the measure of angle BCD?	NA
	<image>Question: In the given circle with center O, if angle ADB measures 60 degrees and point D lies on the major arc AB, what is the measure of angle BAC?	NA
	<image>In the given diagram, quadrilateral ABCD is inscribed in circle O. If angle A measures 70°, what is the degree measure of angle BCD?	NA
	<image>Question: 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 is the measure of angle ABC?	NA
	<image>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 PAB?	NA
	<image>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 150 degrees, what is the degree measure of angle BAC?	NA
	<image>Question: In the given circle with center O, if angle ADB measures 75° and angle ABC measures 60°, what is the measure of angle BOC?	NA
	<image>Question: If CD is the diameter of circle O and points A and B lie on circle O such that AC = AB, and angle ACD measures 40°, what is the degree measure of angle ABC?	NA
	<image>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 3.0 units, DB measures 1.5 units, and DE measures 2.4 units, what is the length of BC?	NA
	<image>If AB is the diameter of circle O, and C and D are points on circle O with angle ADB measuring 30 degrees, then what is the measure of angle ACB?	NA
	<image>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 50 degrees, what is the measure of angle ACB?	NA
	<image>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 30°, what is the degree measure of angle OCD at point C on the circle in the first quadrant?	NA
	<image>Question: In the given circle with center O, AB is the diameter. Points C and D lie on the circumference. If angle ADB measures 60°, what is the measure of angle ACB?	NA
	<image>Question: 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, and angle ACF is 32 degrees, what is the measure of angle CFE?	NA
	<image>Question: In the given circle with center O, AB is the diameter. Points C and D lie on the circumference. If angle ACD measures 75°, what is the measure of angle BCD?	NA
	<image>If quadrilateral ABCD is inscribed in circle O and AB is the diameter of the circle, with angle AOD measuring 60 degrees and chord AC bisecting angle DAB, what is the measure of angle BCD?	NA
	<image>Question: 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 A if angle C is 46 degrees?	NA
	<image>If quadrilateral ABCD is inscribed in circle O with AB as the diameter, and angle D measures 105°, what is the measure of angle ACB when AC is connected?	NA
	<image>Question: In the given figure, triangle ABC has DE as its median line. If AB has a length of 4.5 units, what is the length of DE?	NA
	<image>Question: In the given diagram, quadrilateral ABCD is circumscribed around circle O. If angle AOB measures 120°, what is the degree measure of angle ADC?	NA
	<image>Question: In the given figure, the extended lines of the chords AB and CD of circle O intersect outside the circle at point E. If angle AOC = 120° and angle BCD = 30°, find the measure of angle E.	NA
	<image>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 50° and angle AEB is 120°, what is the measure of angle ABC?	NA
	<image>Question: In the given diagram, AB is tangent to circle O at point A. Line OB intersects circle O at point C. Point D is on circle O. Connect AD, CD, and OA. If angle B measures 20°, what is the measure of angle ACD?	NA
	<image>Question: In the given circle with center O, points A, B, and C lie on the circumference. If angle AOB measures 120° and angle OBC measures 30°, what is the measure of angle OAC?	NA
	<image>Question: In the given diagram, there is a circle O with chords AB and CD intersecting at point E inside the circle. The tangent passing through point C intersects the extension of OB at point F. If angle A measures 45 degrees and angle D measures 29 degrees, what is the measure of angle F?	NA
	<image>Question: AB is tangent to circle O at point A. Connect OA and OB. If angle AOB measures 50°, what is the measure of angle B?	NA
	<image>Question: In the given circle with center O, quadrilateral ABCD is inscribed such that AB = CD and A is the midpoint of BD. If angle BDC measures 60°, what is the measure of angle ADB?	NA
	<image>Question: In the given diagram, points A, B, and C lie on circle O. If the measure of angle BAC is 40° and a tangent line to circle O passing through point C intersects the extension of OA at point D, what is the measure of angle D?	NA
	<image>In the given figure of triangle DABC, point D lies on side BC. It is given that angle DAC measures 45.0 degrees and angle DAB measures 60.0 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?	NA
	<image>Question: In circle O, if AB is equal to CB and the line segment AD is parallel to BC, with the tangent at point B intersecting line AD at point D, what is the measure of angle ADB if angle ABC equals 52 degrees?	NA
	<image>In the given diagram with circle O, if angle ADB measures 30 degrees and the tangent to circle O passing through point D intersects the extension of AB at point E, what is the degree measure of angle E?	NA
	<image>Question: 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, what is the measure of angle BDC given that angle CAB measures 35 degrees?	NA
	<image>Question: In the given circle with center O, AB is the diameter, and chord CD is perpendicular to AB at point E. If angle OCE measures 40°, what is the measure of angle ACD?	NA
	<image>Question: In the provided figure, if AB is the diameter of circle O and point C lies on circle O, and angle B measures 36°, what is the degree measure of arc AC?	NA
	<image>In the given diagram, with AD as the diameter of the circumcircle circle O of triangle ABC, and angle CAD measuring 42°, what is the measure of angle ACB?	NA
	<image>Question: If line segment BM is tangent to circle O at point B and angle ABC is 70°, what is the degree measure of angle ACB?	NA
	<image>Question: In the figure provided, circle O contains two perpendicular chords, AB and CD. If angle ACD measures 40°, what is the degree measure of angle BCD?	NA
	<image>Question: 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 AOB measures 60 degrees, what is the measure of angle BCD?	NA
	<image>Question: If points A and D are on the major and minor arcs BC respectively, and angle BOC measures 110°, what is the measure of angle BDC?	NA
	<image>Question: During the same night, Xiaoming walked from point B towards point C under the street light A. The length of the shadow CD was measured as 1.2 units. When he continued walking towards point D, the end E of the shadow DE was measured, and it was found that the angle of elevation from Xiaoming to the street light A was 30.0 degrees. Given that Xiaoming's height is 1.8 units, what is the height AB of the street light A?	NA
	<image>Question: If points A, B, and C lie on circle O and point D is located on the extension of line AB, with angle AOC measuring 130 degrees, what is the measure of angle CBD?	NA
	<image>Question: In the given illustration, there is an equilateral triangle ABC inscribed within circle O. If triangle ABC is rotated 45° clockwise around the center O, forming triangle DEF, and then CF is connected, what is the measure of angle CFE?	NA
	<image>If point B lies on AC in circle O and the measure of angle AOC is 80 degrees, what is the degree measure of angle ABC?	NA
	<image>Question: In the given semicircle with AB as its diameter, CD is a chord parallel to AB. If angle ACD measures 30°, what is the measure of angle B?	NA
	<image>In the given figure, circle O has diameter AB, and points C and D are the trisection points of BE. If angle AOE measures 45°, what is the degree measure of angle BOD?	NA
	<image>Question: If BD is the diameter of circle O, with points A and C lying on circle O and angle BCD measuring 60 degrees, what is the measure of angle A?	NA
	<image>Question: If points A, B, C, D, and E lie on circle O, with angle BAC measuring 25 degrees and angle CED measuring 45 degrees, what is the measure of angle BOD?	NA
	<image>In the given diagram, if points A, B, and C lie on circle O and angle AOC is 120 degrees, what is the measurement of angle ABC?	NA
	<image>Question: If angle AOB measures 120 degrees and points A, B, and C lie on circle O, what is the measure of angle ACB?	NA
	<image>In the given diagram, circle O has AB as its diameter and CD as a chord. If angle ACD measures 36°, what is the measure of angle BCD?	NA
	<image>Question: 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 BCD if angle ADB is 72°?	NA
	<image>If A, B, and C are all points on circle O, and angle ACB measures 30 degrees, what is the measure of angle AOB?	NA
	<image>If AB is the diameter of circle O and point D is located on circle O, with angle BDC measuring 25°, what is the measure of angle AOC?	NA
	<image>Question: In the given circle with center O, AB is the diameter. If angle BCD measures 30°, what is the measure of angle BAD?	NA
	<image>Question: In the given diagram, chords AB and CD intersect at point P inside circle O. If angle A measures 40° and angle C measures 60°, what is the degree measure of angle APD?	NA
	<image>Question: 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 42°, what is the degree measure of angle D?	NA
	<image>In the given diagram, circle O has diameter AB and points C and D lie on the circle. If angle ACB measures 60 degrees, what is the degree measure of angle ADB?	NA
	<image>Question: In the given figure, AB and EG are the diameters of circle O. EF and EB are chords of circle O, with EF = EB. EF intersects AB at point C. Connect OF. If angle EOB = 30°, find the measure of angle F.	NA
	<image>Question: 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, what is the relationship between the measures of angles A and C?	NA
	<image>Question: In the given diagram, if AB is the diameter of circle O and points C and D lie on the circle, with angle DCB measuring 45 degrees, what is the measurement of angle AOD?	NA
	<image>If angle BAO measures 45 degrees in the given diagram where points A, B, and C lie on circle O, what is the measure of angle ACB?	NA
	<image>Question: In the given figure, AB is the diameter of circle O, and CD is a chord of circle O. Connect AC, AD, and BD. If angle ADB measures 70°, what is the measure of angle BAC?	NA
	<image>Question: If angle BOC measures 108° in the diagram with points A, B, and C on circle O, what is the measure of angle BAC?	NA
	<image>Question: If angle OBC is 30 degrees and points A, B, and C lie on the circle with center O, what is the measure of angle ACB?	NA
	<image>Question: In the given circle with center O, points A, B, and C lie on the circumference. The line segment CO extends to intersect AB at point D. If angle A measures 40° and angle ADC measures 120°, what is the measure of angle B?	NA
	<image>If the measure of angle ACB is 36 degrees in the given figure where points A, B, and C lie on circle O, what is the degree measure of angle AOB?	NA
	<image>Question: In the given diagram, quadrilateral ABCD is inscribed in circle O, with BC as the diameter. If angle ADB measures 42° and D is the midpoint of AC, what is the measure of angle BAC?	NA
	<image>Question: In the given circle with center O, AB is the diameter. Points C and D lie on the circumference. If angle ADB measures 64°, what is the measure of angle ACB?	NA
	<image>Question: In the given illustration, if angle ACB in circle O measures 60° and point P lies on minor arc AB with angle AOP measuring 30°, what is the degree measure of angle PBO?	NA
	<image>Question: In the given diagram, quadrilateral ABCD is inscribed in circle O. If angle B measures 120°, what is the degree measure of angle ADC?	NA
	<image>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 BDC?	NA
	<image>Question: In the given illustration, if AB is the diameter of circle O and points C and D lie on the circle, with angle BOC measuring 108°, what is the measure of angle CDB?	NA
	<image>Question: In the given circle with center O, AB is the diameter. Points C and D lie on the circumference. If angle ADB measures 35°, what is the measure of angle ACB?	NA

<image>Question: In the given diagram, square ABCD is enclosed within circle O, and point P is located on the side AD. What is the measure of angle BPC?

NA

<image>Question: In the given figure, AB is the diameter of the semicircle O. Point C lies on the semicircle O, distinct from points A and B. D is the midpoint of AC. Extend DC to intersect the extension line of AB at point E. If angle CAE measures 20°, what is the measure of angle E?

NA

<image>In the given diagram of circle O, where AB is the diameter and CD is a chord, if angle BCD measures 60°, what is the degree measure of angle CAD?

NA

<image>Question: In the given circle with center O, points A, B, and C lie on the circumference. If angle AOB measures 70° and angle BOC measures 40°, what is the measure of angle ACB?

NA

<image>Question: Given that AB is the diameter of circle O and points C and D lie on circle O, with angle ACD measuring 30 degrees, what is the measure of angle DOB?

NA

<image>Question: In the same diagram, AB is the diameter of circle O, and C, D are two points on the circle. Given that angle ADB = 60°, find the measure of angle BOC.

NA

<image>Question: In the given illustration, circle O has diameter AB, and chords AC and BC. If angle C measures 40°, what is the measure of angle A?

NA

<image>Question: In the given diagram, points A, B, and C lie on circle O. If the measure of angle AOB is 120 degrees, what is the measure of angle ACB?

NA

<image>If point O is the incenter of triangle ABC and angle A measures 50 degrees, what is the measure of angle BOC when BO and CO are connected?

NA

<image>If quadrilateral ABCD is inscribed in circle O and angle B measures 60 degrees, what is the degree measure of angle D?

NA

<image>Question: In the given diagram, triangle ABC is inscribed in circle O, with CD as the diameter of circle O. If angle BAC = 30°, find the measure of angle DBC.

NA

<image>Question: In the given figure, circle O is the circumcircle of equilateral triangle ABC. Point D is a point on arc BC, and angle BAC = 30°. What is the measure of angle ACD?

NA

<image>Question: If circle O is the circumcircle of triangle ABC, angle A measures 60 degrees, and E is the midpoint of side BC, what is the measure of angle BCD when OE is extended to intersect circle O at point D and BD is connected?

NA

<image>Question: In the given diagram, circle O has diameter AB, and points C and D lie on circle O. If angle ADB measures 52°, what is the degree measure of angle ACB?

NA

<image>Question: 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 AOB if angle B measures 52 degrees?

NA

<image>Question: If points A, B, and C lie on circle O and angle AOB measures 128°, what is the degree measure of angle ACB?

NA

<image>Question: 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 ADE if angle BAD is 45 degrees?

NA

<image>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 ACD measures 25°, what is the degree measure of angle ADC?

NA

<image>Question: If AB is the diameter of circle O, 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, what is the measure of angle ACD?

NA

<image>Question: In the given diagram, AB is the diameter of circle O. EF and EB are chords of circle O. If angle AOF measures 50°, what is the measure of angle E?

NA

<image>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 BAC is 60 degrees, what is the measure of angle OPC?

NA

<image>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 BDC?

NA

<image>Question: 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 BCD measuring 40°, what is the measure of angle BAC?

NA

<image>Question: If point Q lies on the circumcircle of regular hexagon ABCDEF, which is circle O, and Q is not coincident with A or B, what is the measure of angle AQD?

NA

<image>Question: 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 124 degrees, what is the measure of angle P?

NA

<image>Question: 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 45°, what is the measure of angle D?

NA

<image>Question: If BC is the diameter of circle O, and points A and D lie on circle O, with angle BAC measuring 40°, what is the measure of angle ADC?

NA

<image>Question: In the given circle with center O, AB is the diameter. Point C lies on the circumference, and D is the midpoint of arc AC. If angle AOC measures 120°, what is the measure of angle BCD?

NA

<image>Question: In the given circle with center O, if angle ADB measures 60 degrees and point D lies on the major arc AB, what is the measure of angle BAC?

NA

<image>In the given diagram, quadrilateral ABCD is inscribed in circle O. If angle A measures 70°, what is the degree measure of angle BCD?

NA

<image>Question: 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 is the measure of angle ABC?

NA

<image>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 PAB?

NA

<image>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 150 degrees, what is the degree measure of angle BAC?

NA

<image>Question: In the given circle with center O, if angle ADB measures 75° and angle ABC measures 60°, what is the measure of angle BOC?

NA

<image>Question: If CD is the diameter of circle O and points A and B lie on circle O such that AC = AB, and angle ACD measures 40°, what is the degree measure of angle ABC?

NA

<image>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 3.0 units, DB measures 1.5 units, and DE measures 2.4 units, what is the length of BC?

NA

<image>If AB is the diameter of circle O, and C and D are points on circle O with angle ADB measuring 30 degrees, then what is the measure of angle ACB?

NA

<image>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 50 degrees, what is the measure of angle ACB?

NA

<image>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 30°, what is the degree measure of angle OCD at point C on the circle in the first quadrant?

NA

<image>Question: In the given circle with center O, AB is the diameter. Points C and D lie on the circumference. If angle ADB measures 60°, what is the measure of angle ACB?

NA

<image>Question: 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, and angle ACF is 32 degrees, what is the measure of angle CFE?

NA

<image>Question: In the given circle with center O, AB is the diameter. Points C and D lie on the circumference. If angle ACD measures 75°, what is the measure of angle BCD?

NA

<image>If quadrilateral ABCD is inscribed in circle O and AB is the diameter of the circle, with angle AOD measuring 60 degrees and chord AC bisecting angle DAB, what is the measure of angle BCD?

NA

<image>Question: 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 A if angle C is 46 degrees?

NA

<image>If quadrilateral ABCD is inscribed in circle O with AB as the diameter, and angle D measures 105°, what is the measure of angle ACB when AC is connected?

NA

<image>Question: In the given figure, triangle ABC has DE as its median line. If AB has a length of 4.5 units, what is the length of DE?

NA

<image>Question: In the given diagram, quadrilateral ABCD is circumscribed around circle O. If angle AOB measures 120°, what is the degree measure of angle ADC?

NA

<image>Question: In the given figure, the extended lines of the chords AB and CD of circle O intersect outside the circle at point E. If angle AOC = 120° and angle BCD = 30°, find the measure of angle E.

NA

<image>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 50° and angle AEB is 120°, what is the measure of angle ABC?

NA

<image>Question: In the given diagram, AB is tangent to circle O at point A. Line OB intersects circle O at point C. Point D is on circle O. Connect AD, CD, and OA. If angle B measures 20°, what is the measure of angle ACD?

NA

<image>Question: In the given circle with center O, points A, B, and C lie on the circumference. If angle AOB measures 120° and angle OBC measures 30°, what is the measure of angle OAC?

NA

<image>Question: In the given diagram, there is a circle O with chords AB and CD intersecting at point E inside the circle. The tangent passing through point C intersects the extension of OB at point F. If angle A measures 45 degrees and angle D measures 29 degrees, what is the measure of angle F?

NA

<image>Question: AB is tangent to circle O at point A. Connect OA and OB. If angle AOB measures 50°, what is the measure of angle B?

NA

<image>Question: In the given circle with center O, quadrilateral ABCD is inscribed such that AB = CD and A is the midpoint of BD. If angle BDC measures 60°, what is the measure of angle ADB?

NA

<image>Question: In the given diagram, points A, B, and C lie on circle O. If the measure of angle BAC is 40° and a tangent line to circle O passing through point C intersects the extension of OA at point D, what is the measure of angle D?

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<image>In the given figure of triangle DABC, point D lies on side BC. It is given that angle DAC measures 45.0 degrees and angle DAB measures 60.0 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?

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<image>Question: In circle O, if AB is equal to CB and the line segment AD is parallel to BC, with the tangent at point B intersecting line AD at point D, what is the measure of angle ADB if angle ABC equals 52 degrees?

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<image>In the given diagram with circle O, if angle ADB measures 30 degrees and the tangent to circle O passing through point D intersects the extension of AB at point E, what is the degree measure of angle E?

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<image>Question: 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, what is the measure of angle BDC given that angle CAB measures 35 degrees?

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<image>Question: In the given circle with center O, AB is the diameter, and chord CD is perpendicular to AB at point E. If angle OCE measures 40°, what is the measure of angle ACD?

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<image>Question: In the provided figure, if AB is the diameter of circle O and point C lies on circle O, and angle B measures 36°, what is the degree measure of arc AC?

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<image>In the given diagram, with AD as the diameter of the circumcircle circle O of triangle ABC, and angle CAD measuring 42°, what is the measure of angle ACB?

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<image>Question: If line segment BM is tangent to circle O at point B and angle ABC is 70°, what is the degree measure of angle ACB?

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<image>Question: In the figure provided, circle O contains two perpendicular chords, AB and CD. If angle ACD measures 40°, what is the degree measure of angle BCD?

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<image>Question: 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 AOB measures 60 degrees, what is the measure of angle BCD?

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<image>Question: If points A and D are on the major and minor arcs BC respectively, and angle BOC measures 110°, what is the measure of angle BDC?

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<image>Question: During the same night, Xiaoming walked from point B towards point C under the street light A. The length of the shadow CD was measured as 1.2 units. When he continued walking towards point D, the end E of the shadow DE was measured, and it was found that the angle of elevation from Xiaoming to the street light A was 30.0 degrees. Given that Xiaoming's height is 1.8 units, what is the height AB of the street light A?

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<image>Question: If points A, B, and C lie on circle O and point D is located on the extension of line AB, with angle AOC measuring 130 degrees, what is the measure of angle CBD?

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<image>Question: In the given illustration, there is an equilateral triangle ABC inscribed within circle O. If triangle ABC is rotated 45° clockwise around the center O, forming triangle DEF, and then CF is connected, what is the measure of angle CFE?

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<image>If point B lies on AC in circle O and the measure of angle AOC is 80 degrees, what is the degree measure of angle ABC?

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<image>Question: In the given semicircle with AB as its diameter, CD is a chord parallel to AB. If angle ACD measures 30°, what is the measure of angle B?

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<image>In the given figure, circle O has diameter AB, and points C and D are the trisection points of BE. If angle AOE measures 45°, what is the degree measure of angle BOD?

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<image>Question: If BD is the diameter of circle O, with points A and C lying on circle O and angle BCD measuring 60 degrees, what is the measure of angle A?

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<image>Question: If points A, B, C, D, and E lie on circle O, with angle BAC measuring 25 degrees and angle CED measuring 45 degrees, what is the measure of angle BOD?

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<image>In the given diagram, if points A, B, and C lie on circle O and angle AOC is 120 degrees, what is the measurement of angle ABC?

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<image>Question: If angle AOB measures 120 degrees and points A, B, and C lie on circle O, what is the measure of angle ACB?

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<image>In the given diagram, circle O has AB as its diameter and CD as a chord. If angle ACD measures 36°, what is the measure of angle BCD?

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<image>Question: 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 BCD if angle ADB is 72°?

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

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<image>If AB is the diameter of circle O and point D is located on circle O, with angle BDC measuring 25°, what is the measure of angle AOC?

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<image>Question: In the given circle with center O, AB is the diameter. If angle BCD measures 30°, what is the measure of angle BAD?

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<image>Question: In the given diagram, chords AB and CD intersect at point P inside circle O. If angle A measures 40° and angle C measures 60°, what is the degree measure of angle APD?

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<image>Question: 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 42°, what is the degree measure of angle D?

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<image>In the given diagram, circle O has diameter AB and points C and D lie on the circle. If angle ACB measures 60 degrees, what is the degree measure of angle ADB?

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<image>Question: In the given figure, AB and EG are the diameters of circle O. EF and EB are chords of circle O, with EF = EB. EF intersects AB at point C. Connect OF. If angle EOB = 30°, find the measure of angle F.

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<image>Question: 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, what is the relationship between the measures of angles A and C?

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<image>Question: In the given diagram, if AB is the diameter of circle O and points C and D lie on the circle, with angle DCB measuring 45 degrees, what is the measurement of angle AOD?

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<image>If angle BAO measures 45 degrees in the given diagram where points A, B, and C lie on circle O, what is the measure of angle ACB?

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<image>Question: In the given figure, AB is the diameter of circle O, and CD is a chord of circle O. Connect AC, AD, and BD. If angle ADB measures 70°, what is the measure of angle BAC?

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<image>Question: If angle BOC measures 108° in the diagram with points A, B, and C on circle O, what is the measure of angle BAC?

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<image>Question: If angle OBC is 30 degrees and points A, B, and C lie on the circle with center O, what is the measure of angle ACB?

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<image>Question: In the given circle with center O, points A, B, and C lie on the circumference. The line segment CO extends to intersect AB at point D. If angle A measures 40° and angle ADC measures 120°, what is the measure of angle B?

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<image>If the measure of angle ACB is 36 degrees in the given figure where points A, B, and C lie on circle O, what is the degree measure of angle AOB?

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<image>Question: In the given diagram, quadrilateral ABCD is inscribed in circle O, with BC as the diameter. If angle ADB measures 42° and D is the midpoint of AC, what is the measure of angle BAC?

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<image>Question: In the given circle with center O, AB is the diameter. Points C and D lie on the circumference. If angle ADB measures 64°, what is the measure of angle ACB?

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<image>Question: In the given illustration, if angle ACB in circle O measures 60° and point P lies on minor arc AB with angle AOP measuring 30°, what is the degree measure of angle PBO?

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<image>Question: In the given diagram, quadrilateral ABCD is inscribed in circle O. If angle B measures 120°, what is the degree measure of angle ADC?

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<image>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 BDC?

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<image>Question: In the given illustration, if AB is the diameter of circle O and points C and D lie on the circle, with angle BOC measuring 108°, what is the measure of angle CDB?

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<image>Question: In the given circle with center O, AB is the diameter. Points C and D lie on the circumference. If angle ADB measures 35°, what is the measure of angle ACB?

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