problem
stringlengths 7
719
| answer
stringlengths 0
22
| images
images listlengths 1
1
|
|---|---|---|
<image>In the given diagram, quadrilateral ABCD is circumscribed in circle O, and the measure of angle A is 125 degrees. What is the degree measure of angle C? | 55° | |
<image>What is the minimum length of the carpet required if the height is 3.0 and the slope length is 5.0 stair steps? | 7米 | |
<image>Given the diagram, triangle ABC is circumscribed by circle O. Connect OA and OB. If angle ABO = 35°, then the degree of angle C is (). | 55° | |
<image>In the given illustration, points A, B, C, and D lie on circle O, with point O as the center. The quadrilateral OBCD is formed by connecting points A, B, C, and D on the circle. If OD is equal to BC and OB is equal to CD, what is the measure of angle A? | 30° | |
<image>If quadrilateral ABCD is inscribed in circle O such that angle BAD equals angle BCD and both measure 90 degrees, and AD equals CD, while angle ADC measures 120 degrees, and E is the midpoint of arc BC, what is the measure of angle CDE if DE is connected? | 30° | |
<image>As shown in the figure, AB is the diameter of circle O, and points C, D, E are on circle O. If angle BCD = 100°, then the measure of angle AED is (). | 10° | |
<image>Given that AB is the diameter of semicircle O and points C and D trisect AE. If the measure of angle COD is 50 degrees, what is the degree measure of angle BOE? | 30° | |
<image>As shown in the figure, AB is the diameter of circle O, and points C and D are on circle O. If angle ABC = 60°, then the degree measure of angle D is () | 30° | |
<image>If points A, B, and C lie on circle O and angle BOC measures 130 degrees, what is the measure of angle A? | 65° | |
<image>What is the measure of angle AOB in circle O, where chord AC intersects radius OB at point D, and angle B measures 60 degrees while angle ADB measures 116 degrees? | 112° | |
<image>What is the measure of angle BOD in circle O, where segment AB serves as the diameter, and chord CD is perpendicular to AB, with angle CAB measuring 25 degrees? | 50° | |
<image>In the given diagram, if AB is the diameter of circle O, and point C lies on circle O, with angle A measuring 24 degrees, what is the measure of angle B? | 66° | |
<image>If angle 1 measures 20.0 degrees and the ruler and triangle ruler with a 30.0 degree angle are placed together as shown in the diagram, what is the measure of angle 2? | 50° | |
<image>In the given diagram, circle O contains points A, B, and C. If BC is parallel to OA and angle A measures 20°, what is the measure of angle B? | 40° | |
<image>In the given diagram, circle O has diameter AB with points C and D lying on the circle. If angle D measures 110°, what is the degree measure of angle BAC? | 20° | |
<image>In the given diagram of circle O, where OA and OB represent the radii and AB, AC, and BC are chords, if the measure of angle OAB is 70 degrees, what is the degree measure of angle C? | 20° | |
<image>In the given figure, circle O has AB as its diameter, and triangle ABC is inscribed in the circle. D is a point on AB such that BC=BD. If CD is extended to intersect circle O at point E, and AE is connected, what is the measure of angle CAE if angle B is 70 degrees? | 75° | |
<image>What is the measure of angle B if points A, B, and C lie on circle O, AB is parallel to OC, and angle A is 70 degrees? | 125° | |
<image>As shown in the diagram, AB is the diameter of circle O, C and D are points on circle O. Angle DOB is 64° and angle D is equal to angle B. The value of angle B is (). | 16° | |
<image>What is the measure of angle EDO if line AC is tangent to circle O at point A, point E is the midpoint of arc AB, and angle CAD measures 20 degrees? | 25° | |
<image>What is the measure of angle D in the given diagram of circle O, where points A, B, and C are connected by line segments AB, AC, OB, and OC, and the tangent line BD intersects the extension of line OC at point D when angle A measures 25 degrees? | 40° | |
<image>Triangle ADC is inscribed in circle O, with BC as the diameter of the circle. Given that the measure of angle A is 66 degrees, what is the measure of angle BCD? | 24° | |
<image>As shown in the diagram, AC is the diameter of circle O, BC is the tangent to circle O, and AB intersects circle O at point D. If angle ABC = 65°, then the degree measure of angle COD is (). | 50° | |
<image>As shown in the figure, PA and PB are tangents to circle O, with points of tangency A and B respectively. The extension of line PO intersects circle O at point C. OA, OB, and BC are then connected. If AO = 2 and OP = 4, the measure of angle C is _______. | 30° | |
<image>In the given illustration, triangle ABC is depicted with angle ACB measuring 90°. A circle O is drawn with AC as the diameter, intersecting AB at point D. If CD equals OD, what is the degree measure of angle B? | 60° | |
<image>In the given diagram, circle O has diameter AB, with points C and D lying on the circle. If angle D measures 55 degrees, what is the measure of angle BOC? | 70° | |
<image>In the given illustration, triangle ABC is contained within circle O, and AD serves as the diameter of circle O. If angle C measures 63 degrees, what is the measure of angle DAB? | 27° | |
<image>In the given diagram, circle O has AB as its diameter, with AC serving as the tangent and A as the point of tangency. Line BC intersects circle O at point D, and a line segment OD is drawn. If angle C measures 46 degrees, what is the measure of angle AOD? | 88° | |
<image>As shown in the diagram, AD is a tangent to circle O with point A as the point of tangency. Line DO intersects circle O at point C, and point B is on circle O. Connect lines AB and CB. If angle D = 36°, then the degree measure of angle ABC is (). | 27° | |
<image>In the given diagram, if AB is the diameter of circle O and CD is a chord such that point C does not coincide with point A or point B and point C and point D are on opposite sides of diameter AB, what is the measure of angle BCD if angle AOD equals 110 degrees? | 35° | |
<image>If AB is the diameter of circle O and point D lies on circle O, and the measure of angle AOC is 120 degrees, what is the measure of angle D? | 30° | |
<image>As shown in the diagram, AB is the diameter of circle O, and CD is a chord of circle O. If angle CAB = 51°, then the degree measurement of angle ADC is (). | 39° | |
<image>In the given diagram, quadrilateral ABCD is inscribed in circle O. If we connect OA and OC, and angle AOC measures 116°, what is the degree measure of angle ADC? | 122° | |
<image>In the given illustration, circle O has diameter AB, with points C and D located on AB. The intersection of AC and BD occurs at point E. If angle BEC measures 58°, what is the degree measure of angle DOC? | 64° | |
<image>In the given diagram, circle O has AB as its diameter and CD as a chord. If angle CDB measures 26°, what is the measure of angle ABC? | 64° | |
<image>In the given diagram, circle O has diameter AB. A tangent line is drawn from point C on the circle to intersect the extension of AB at point D. If angle A measures 35 degrees, what is the measure of angle D? | 20° | |
<image>In the given illustration, circle O has AB as the diameter, with points C and D positioned on the circle. C serves as the midpoint of arc BAD. When connecting CD, CB, and OD, CD intersects with AB at point F. If angle AOD is 100 degrees, what is the measure of angle ABC? | 20° | |
<image>As shown in the figure, points A, B, and C are all on circle O. If angle ABO = 20° and angle ACO = 30°, then angle BOC = (). | 100° | |
<image>In the given diagram, in circle O, the tangent CD touches the circle at point D. The line CO intersects circle O at points B and A, where angle A measures 15 degrees. What is the degree measure of angle C? | 60° | |
<image>In the given illustration, circle O has chord AB, and point C is situated on the tangent passing through point B. OC is perpendicular to OA, and it intersects AB at point P. If angle OAB measures 22°, what is the measure of angle OCB? | 44° | |
<image>As shown in the figure, triangle ABC is inscribed in circle O, AD is the diameter of circle O. If angle C = 63°, then angle DAB is equal to () | 27° | |
<image>In the given semicircle O with diameter AB, where AC equals BC, and D is a point on AC, how much is the measure of angle DOC if angle DEC measures 65 degrees and BD intersects OC at point E? | 40° | |
<image>In the given illustration, point P is located outside circle O, and tangent lines PA and PB intersect the circle at points A and B respectively. Point C lies to the left of circle O, above line AB. If angle P measures 50°, what is the measure of angle ACB? | 65° | |
<image>As shown in the figure, AB is the diameter of circle O. Point C is a point outside circle O. CA and CD are tangent to circle O at points A and D respectively. Connect BD and AD. If angle ACD = 50°, then the degree of angle DBA is (). | 65° | |
<image>In the given diagram, circle O has diameter AB, with point C lying on the circle. A tangent passing through C intersects the extended line of AB at E, and point D lies on AC but does not coincide with points A and C. If angle D measures 110°, what is the degree measure of angle AEC? | 50° | |
<image>What is the degree measure of angle BCD in quadrilateral ABCD, where the quadrilateral is inscribed in circle O and angle BOD measures 108 degrees according to the given diagram? | 126° | |
<image>Given: In the diagram, OA and OB are two radii of circle O, and angle AOB = 100°. Point C is on circle O. The degree measure of angle ACB is (). | 50° | |
<image>In the given diagram, quadrilateral ABDE is inscribed in circle O, with CE as the diameter of the circle. Connecting BC and DC in the quadrilateral, if angle BDC measures 20°, what is the measure of angle A? | 110° | |
<image>In the given diagram, circle O has diameter AB, with PA as a tangent at point A. Line segment PO intersects circle O at point C, and BC is connected. If angle P measures 40 degrees, what is the measure of angle B? | 25° | |
<image>If AB is the diameter of circle O, and points C and D lie on circle O with angle D measuring 56°, what is the measure of angle ABC? | 34° | |
<image>If AB is the diameter of circle O and points C, D, and E are located on circle O, with angle AED measuring 25°, what is the measurement of angle BCD? | 115° | |
<image>In the provided figure, there is a triangular fragment ABC with a missing corner. If angle A measures 40.0 degrees and angle B measures 65.0 degrees, what is the degree measure of angle C before the triangle is broken? | 75° | |
<image>If point A, B, and C are all located on circle O, and angle AOB measures 72 degrees, what is the measure of angle ACB? | 36° | |
<image>In the given diagram, it is depicted that circle O has diameter AB, where point C is the midpoint of the arc, and point D lies on the circle O on the opposite side of AB. What is the measurement of angle D? | 45° | |
<image>As shown in the figure, AB is the diameter of circle O. Chord CD is perpendicular to AB, and DE is parallel to CB. DE intersects circle O at point E. If angle CBA = 20°, then the degree measure of angle AOE is (). | 100° | |
<image>In the given diagram of circle O, if angle OAC plus angle C equals 50 degrees, what is the measure of angle BAC? | 40° | |
<image>What is the minimum length of DE if point D is a moving point on the line segment BC of equilateral triangle ABC with a side length of 2 and circle A with a radius of 1, and a tangent line to circle A is drawn through point D to the point of tangency E? | √2 | |
<image>In the given diagram, circle O has diameter AB, with points C and D lying on the circle. If angle AOD measures 30 degrees, what is the measure of angle BCD? | 105° | |
<image>In the given diagram, line BC is a tangent to circle O at point B. Line CO intersects and extends circle O at point A, and AB is connected. If angle BAC measures 36°, what is the measure of angle C? | 18° | |
<image>In the given diagram, circle O has diameter AB, and AP is a tangent to circle O with A as the point of tangency. If angle BAC measures 58°, what is the measure of angle P? | 58° | |
<image>In the given diagram, points A and B are points of tangency for the circle O with the lines PA and PB, respectively, and the length of PA is equal to the length of AB. What is the measure of angle AOB? | 120° | |
<image>In the given diagram of the cyclic quadrilateral ABDE within circle O, where AB is the diameter, and angle E measures 106°, if the tangent line DC passing through point D intersects the extension of AB at point C, what is the measure of angle C? | 58° | |
<image>As shown in the figure, AB is the diameter of the circle O. Point E and C are on the circle O. Point A is the midpoint of EC. Draw a tangent to circle O passing through point A, intersecting the extension of BC at point D. Connect EC. If angle ADB = 58.5°, then the degree of angle ACE is () | 31.5° | |
<image>What is the measure of angle D if line AB is the diameter of circle O, line DA is tangent to circle O at point A, and angle ABC measures 21 degrees? | 48° | |
<image>In the given diagram, triangle ABC is inscribed within circle O, and D is the midpoint of side BC. If OD is extended to intersect circle O at point E, and EC is then connected, what is the measure of angle A if angle OEC is 65°? | 50° | |
<image>In the triangle ABC shown in the diagram, AB = 6. A circle with center A and a radius of 3 is tangent to the side BC at point D. It intersects AC and AB at points E and G, respectively. Point F is a point on arc GE. angle CDE = 18°. The measure of angle GFE is () | 48° | |
<image>What is the measure of angle P if triangle ABC is inscribed in circle O, with point P lying on the minor arc BC, and lines PB and PC are connected, while AB = BC and AC is the diameter? | 135 | |
<image>In the given diagram, chord AB of circle O is both perpendicular to and bisects radius OC. Furthermore, AE is a tangent to circle O and intersects the extension of OC at point E. What is the measure of angle CAE? | 30° | |
<image>What is the measure of angle D if points A, B, and C lie on circle O, and angle ABC is 28 degrees? | 34° | |
<image>In the given diagram, circle O is intersected by the extension of line AO at point C, and line AB is a tangent to the circle at point B. If the measure of angle A is 26 degrees, what is the measure of angle C? | 32° | |
<image>In the provided figure, circle O has diameter AB and chord CD is equal to AB. If angle CAB is measured at 20 degrees, what is the measure of angle BOD? | 40° | |
<image>In the given illustration, lines PA and PB are tangent to circle O at points A and B. If point C lies on circle O and the measure of angle C is 70 degrees, what is the measure of angle P? | 40° | |
<image>In the given diagram, triangle ABC is inscribed in circle O, with angle ACB measuring 40° and AC being equal to BC. The midpoint of BC is E. If we connect OE and extend it to intersect circle O at point D, what is the measure of angle ADO? | 15° | |
<image>In the given diagram, triangle ABC is inscribed within circle O, where AB is equal to AC. If the extension of BO intersects AC at point D and angle ABD measures 23 degrees, what is the measure of angle A? | 46° | |
<image>In the given diagram, circle O has line segment AB as its diameter and CD as a chord. A tangent passing through point C intersects the extension of AB at point E, and angle E measures 42°. What is the measure of angle CDB? | 24° | |
<image>In the given diagram, circle O has AB as its diameter. The line PA is tangent to circle O at point A, and PO intersects circle O at point C. If angle BCO measures 10 degrees, what is the measure of angle P? | 70° | |
<image>In the given diagram, if AB is the diameter of circle O and point C lies on the extension of AB, with CD being tangent to circle O at point D, what is the measure of angle C if angle CDA is 118°? | 34° | |
<image>As shown in the figure, PA is a tangent to the circle O. Connect OP and OA. If angle A = 50°, then the measure of angle POA is ( ). | 40° | |
<image>Given triangle ABC circumscribes circle O, connect AO and extend it to intersect BC at point D. If angle C = 50°, then the degree of angle BAD is () | 40° | |
<image>In the given diagram, triangle ABG is circumscribed in circle O. When AO is connected and extended to intersect with BC at point D, and if angle B is 70 degrees and angle C is 50 degrees, what is the degree measure of angle ADB? | 70° | |
<image>In the provided figure, where ABCD is a quadrilateral with a circle inscribed, if angle C measures 80.0 degrees, what is the measure of angle A? | 100° | |
<image>In the given diagram, points P, A, and B are positioned with PA and PB being tangents to circle O at points A and B respectively. If angle P measures 70 degrees, what is the measure of angle C? | 55° | |
<image>In the given illustration, if AB is the diameter of circle O, AD is a tangent to circle O, and BD intersects circle O at point C, what is the measure of angle B if angle CAD is 50 degrees? | 50° | |
<image>In the given diagram, circle O has diameter AB and tangent MN with point of tangency at N. If angle MNB measures 52°, what is the measurement of angle NOA? | 76° | |
<image>What is the measure of angle P if PA and PB are tangents to circle O, A and B are the points of tangency, AC is the diameter of the circle, and angle BAC measures 25.5 degrees? | 51° | |
<image>In the given diagram of triangle ABC, where AB is equal to AC and angle ABC measures 70 degrees, and O is the incenter of the triangle, what is the measure of angle BOC? | 110° | |
<image>In the given illustration, points PA and PB are tangents to circle O at points A and B respectively. If point C lies on circle O and lines AC and BC are connected, what is the measure of angle ACB if angle P is 50 degrees? | 115° | |
<image>In the given diagram, line AB is tangent to circle O at point A, and line OB intersects circle O at point C. Point D lies on circle O. If angle ADC measures 28 degrees, what is the measure of angle ABO? | 34° | |
<image>As shown in the diagram, circle O is the circumcircle of triangle ABC. Connecting OB, if angle OBC = 30°, then the degree of angle A is (). | 60° | |
<image>As shown in the picture, circle O is the circumcircle of triangle ABC with a radius of 3cm. If BC = 3cm, the degree measure of angle A is (). | 30° | |
<image>As shown in the diagram, points A and B are both on circle O. Line PC is tangent to circle O at point C. If angle CAP = 35°, then the size of angle APC is () | 20° | |
<image>As shown in the diagram, PA and PB are tangent to the circle O at points A and B respectively. Point C is a point on the circle O. Connect AC and BC. If angle P = 78°, then the degree of angle ACB is () | 51° | |
<image>The following picture shows the incenter of triangle ABC. The line EF, parallel to BC, intersects AB and AC at points E and F, respectively. If angle A is 70°, then the measure of angle BIC is (). | 125° | |
<image>In the given diagram, line PA is a tangent to circle O at point A. The extension of PO intersects circle O at point B. If angle P measures 38 degrees, what is the measure of angle B? | 26° | |
<image>As shown in the figure, the incircle of triangle ABC, denoted by circle O, is tangent to AB, BC, AC at points D, E, F respectively. If angle DEF = 55°, then the degree measure of angle A is (). | 70° | |
<image>In the given diagram of triangle ABC, where angle A is 80 degrees, if point O serves as the incenter of the triangle, what is the measure of angle BOC? | 130° | |
<image>In the given figure, AB is a tangent to circle O, AC is a chord, and CB intersects circle O at point D. If CB passes through the center O and angle ACB measures 28°, what is the measure of angle B? | 34° | |
<image>If triangle ABC is inscribed in circle O, where AC is the diameter of the circle and AB equals BC, and the measure of angle DBC is 32 degrees, what is the measure of angle BCD? | 103° | |
<image>In the illustration, triangle ABC is inscribed in circle O with AB as the diameter, and point D lies on circle O. Given that angle BAC measures 35°, what is the measure of angle ADC? | 55° | |
<image>If triangle ABC is inscribed in circle O, with AB equal to AC, angle ACB measuring 65 degrees, and point C being the midpoint of arc BD, what is the measure of angle ACD? | 15° |
Subsets and Splits