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In the diagram, line AB is parallel to line CD. The angle bisector of angle AOB intersects line CD at point O and forms an angle 2 with line OB. If angle 1 measures 30°, what is the measure of angle 2? then, please answer, In the given diagram, if angle 1 has a measure of 35.0 degrees, what is the measure of angle 2? | <answer> 145° </answer> |
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In the given diagram, triangle ABC is depicted with angle ACB measuring 90 degrees and angle BAC measuring 30 degrees. The line segment AD bisects angle BAC and intersects BC at point D. A perpendicular line DE is drawn from point D to AB, intersecting AB at point E. If the length of DE is 1, what is the length of AB? then, please answer, In the given diagram, consider right triangle ABC where angle C equals 90 degrees. The perpendicular bisector of side AB, labeled as DE, intersects BC at point D and intersects AB at point E. If AD divides angle CAB into two angles and the ratio of angle CAD to angle BAD is 0.4, what is the measure of angle ADC? | <answer> 75° </answer> |
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In the given diagram, there is a triangle ABC where angle C measures 90 degrees and angle B is twice the measure of angle A. The lines l1 and l2 pass through points A and B respectively, intersecting line BC at points D and E. If the ratio of the lengths of segments CD and BE is 3:5, what is the value of tan∠1? then, please answer, As shown in the figure, line l1 is parallel to line l2, and triangle ABC is an equilateral triangle. If angle 1 = 40°, then the size of angle 2 is () | <answer> 80° </answer> |
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In the given figure, triangle ABC is depicted with points D and B serving as the midpoints of sides AC and BC respectively. The line segment BD intersects side AC at point E. If the area of triangle CDE is 12 square units, what is the area of triangle ABE? then, please answer, In the given triangle ABC, if line BD is perpendicular to AC and also bisects it, and angle A measures 20 degrees, what is the measure of angle CBD? | <answer> 70° </answer> |
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In the given diagram, triangle ABC is depicted with points P and Q serving as the midpoints of sides BC and AC respectively. Additionally, line segment AP intersects side BC at point M and line segment AQ intersects side AB at point N. If the area of triangle AMN is 12 square units, what is the area of triangle ABC? then, please answer, In the given diagram, if angle B measures 20° and angle C measures 30°, and MP and QN bisect AB and AC perpendicularly, what is the measure of angle PAQ? | <answer> 80° </answer> |
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In the given diagram, triangle ABC is depicted with angle ACB measuring 90 degrees and angle CAB measuring 30 degrees. Point D lies on the arc AB, where CD is perpendicular to AB. The line segment AD intersects BC at point E. If the length of DE is 2, what is the length of CE? then, please answer, What is the measure of angle ADC in right triangle ABC if angle ACB is 90 degrees, angle A is 52 degrees, and a circle with point B as the center and BC as the radius intersects AB at point D? | <answer> 109° </answer> |
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In the given diagram, triangle ABC is depicted with AD serving as the perpendicular bisector of BC and intersecting AB at point D. Additionally, CE is drawn parallel to AD and intersects AB at point E. The line segment DF is perpendicular to CE at point F. If the length of DE is 2 and the length of CF is 3, what is the length of BD? then, please answer, What is the degree measure of angle CFA in triangle ABC, where AB is equal to AC, angle BAC measures 100°, AD is the median of side BC, CE bisects angle BCA, and intersects AB at point E, and AD intersects CE at point F? | <answer> 110° </answer> |
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In the given figure, triangle ABC is depicted with points D and E serving as the midpoints of sides BC and AD respectively. The line segment DE intersects side AC at point F. If the area of triangle AEF is 10 square units, what is the area of triangle ABC? then, please answer, In the given illustration, triangle ABC is depicted with BA being equal to BC. The line segment DE serves as the perpendicular bisector of side AB, intersecting BC and AB at points D and E, respectively. If AD functions as the angle bisector of angle BAC, what is the degree measure of angle B? | <answer> 36° </answer> |
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In the given diagram, triangle ABC is depicted with angle ACB measuring 90 degrees and angle CAB measuring 30 degrees. Point D lies on side AB such that CD is perpendicular to AB. If the length of CD is 2 units, what is the length of AD? then, please answer, What is the measure of angle A in right triangle ABC, where angle ACB is 90 degrees, and an arc is drawn with point B as the center and BC as the radius, intersecting AB at point D, and angle ACD measures 20 degrees? | <answer> 50° </answer> |
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In the given figure, triangle ABC is depicted with AD serving as the altitude and median of angle BAC. If the length of AB is 10 units and AC is 8 units, what is the length of BC? then, please answer, In the given triangle ABC, where AB is equal to AC, and AD is the median of side BC, with angle B measuring 25 degrees, what is the measure of angle BAD? | <answer> 65° </answer> |
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In the given diagram, circle O is depicted with AB as its diameter and CD as a chord. The line OE is tangent to circle O at point D. Additionally, angle BOC measures 100°. Determine the measure of angle ADE. then, please answer, In the provided figure, the quadrilateral ABCD is inscribed in circle O. If one of its exterior angles, DCE, measures 62.0, what is the measure of angle BOD? | <answer> 124° </answer> |
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As shown in the figure, in triangle ABC, angle A = 30° and angle C = 45°. O is the circumcenter of triangle ABC. OD is perpendicular to BC at point D, OE is perpendicular to AC at point E, and OF is perpendicular to AB at point F. The line segments DF, EF, and OF intersect at point O'. If the perimeter of triangle O'EF is 12, then the length of side BC is ( )
A: 6
B: 8
C: 10
D: 12 then, please answer, If a student utilizes a set square with acute angles measuring 30 degrees and 45 degrees and aligns the right-angled vertices, and angle DEB is 60 degrees, what is the measure of angle AOC? | <answer> 45° </answer> |
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In the given diagram, line AB is parallel to line CD and line OE bisects angle AOC. If angle BOD measures 100°, what is the measure of angle COE? then, please answer, In the given figure, point O is the intersection of AE and CD, with AB being parallel to CD. If OC is equal to OE and angle A measures 50 degrees, what is the measure of angle C? | <answer> 25° </answer> |
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In the given figure, triangle ABC is depicted with points O and D located on its sides AB and AC respectively. The line segment BO intersects side AD at point E. Additionally, the line segment OE is perpendicular to side BC at point F. If the lengths of segments OA, OD, and OF are 2, 3, and 1 respectively, what is the length of side BC? then, please answer, In the given diagram, triangle ABC is equilateral with BO as the median line, and BA extended to point D such that AD is equal to AO. What is the degree measure of angle DOB? | <answer> 120° </answer> |
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In the given diagram, line AB is parallel to line CD. The angle bisector of angle ACB intersects line CD at point C and also intersects line AB at point B. If angle 1 measures 50°, what is the measure of angle 2? then, please answer, If AB is parallel to CD and AC equals BC, and angle 1 measures 70 degrees in the given diagram, what is the degree measure of angle 2? | <answer> 40° </answer> |
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In the given diagram, triangle ABC is depicted with points D and E located on sides BC and AB respectively. The line segment DE is parallel to side AC. Additionally, there is a line segment BD that bisects angle ABC. If the length of AD is 2.0 units and DC is 3.0 units, what is the length of BE? then, please answer, In the given diagram, point D lies on AC, while point E lies on AB. It is also given that AB is equal to AC, and BC is equal to BD, and AD is equal to DE, which also equals BE. What is the measure of angle A? | <answer> 45 </answer> |
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In the given diagram, triangle ABC is depicted with points D and E serving as the midpoints of sides AB and AC respectively. Additionally, line segment DF is parallel to BC and intersects side AB at point F. Furthermore, line segment EF is parallel to AD and intersects side AC at point G. Finally, line segment FG is parallel to CD and intersects side BC at point H. then, please answer, What is the sum of angle A, angle B, angle C, angle D, angle E, and angle F in the given equation? | <answer> 360° </answer> |
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In the given figure, quadrilateral ABCD is depicted with points E and F lying on AD and BC respectively. The line segment EF intersects AB at point G and CD at point H. Additionally, there is a line segment CG that bisects angle DCH and intersects AB at point B. If the length of AB is 10 units and the length of GH is 3 units, what is the length of CF? then, please answer, In the given diagram, triangle ABC is depicted with angle ACB measuring 120 degrees. If square ACDE and regular pentagon BCFGH are constructed with sides AC and BC, what is the measure of angle DCF? | <answer> 42° </answer> |
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The diagram shows a right triangle with one angle measuring 30 degrees and the hypotenuse being 12 units long. The right triangle is placed on top of a rectangle, with the hypotenuse of the right triangle aligned with one side of the rectangle. The other two vertices of the right triangle lie on the other two sides of the rectangle. What is the area of the shaded region in the diagram? then, please answer, When Xiaoming was folding a rectangular piece of paper into a regular pentagon, what is the measure of angle 1 in the figure? | <answer> 72° </answer> |
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In the given figure, there is a regular pentagon ABCDE and a regular hexagon FGHMNK. The line l is parallel to the side BC of the pentagon. The intersection points of line l with sides AB, CD, and EF are labeled as A', B', C', D', E', and F'. Similarly, the intersection points of line l with sides FG, GH, and NH are labeled as G', H', and K'. If the length of side BC is 2 units, what is the perimeter of the hexagon FGHMNK? then, please answer, As shown in the diagram, the sides CD and FG of the regular pentagon ABCDE and the regular hexagon EFGHMN are on the line l. The pentagon is on the left side of the hexagon, and both polygons are on the same side of the line l. Therefore, the size of angle DEF is (). | <answer> 48° </answer> |
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In the given diagram, circle O has diameter AB and chord BC. The line OC intersects chord AC at point D. If angle BOC measures 70°, what is the measure of angle A? then, please answer, Given that points A, B, and C lie on circle O and angle AOB measures 80.0 degrees, what is the measure of angle ACB? | <answer> 40° </answer> |
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In the parallelogram ABCD, points E and F are located on sides AB and CD respectively. The line segments DE and BF intersect at point P. Additionally, EF is perpendicular to BD. If the area of triangle BPF is 12 square units and the area of triangle DEF is 6 square units, what is the area of triangle APE? then, please answer, If quadrilateral ABCD is depicted in the figure, with AD equal to BC and point P being the midpoint of diagonal BD while E and F are the midpoints of AB and CD respectively, what is the measure of angle PEF if angle EPF is 130 degrees? | <answer> 25° </answer> |
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In the given diagram, there is a quadrilateral ABCD with its opposite sides parallel to each other. The line l1 intersects AD and BC at points E and F respectively. The line l2 intersects CD and AB at points G and H respectively. If angle 1 measures 50° and angle 2 measures 60°, what is the measure of angle A? then, please answer, As shown in the figure, pentagon ABCDE is a regular pentagon. If l1parallel l2, the value of angle 1 - angle 2 is () | <answer> 72° </answer> |
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In the parallelogram ABCD, the diagonals AC and BD intersect at point E. The line segment AE is extended to intersect the extension of DC at point F. The line segment BE intersects the extension of AD at point G. If the area of triangle ABE is 12 square units, what is the area of triangle DCF? then, please answer, In the given parallelogram ABCD, where angle B measures 60 degrees, the angle bisector of angle BAD intersects BC at point E. If the measure of angle AED is 80 degrees, what is the degree measure of angle EAC? | <answer> 20° </answer> |
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In the given diagram, triangle ABC is depicted with points D and E serving as the midpoints of sides AB and AC respectively. The line segment DE is parallel to BC. If the area of triangle ADE is 10 square units, what is the area of triangle ABC? then, please answer, If points D and E are the midpoints of the sides AB and AC of triangle ABC and connected, and angle B measures 50 degrees while angle A measures 60 degrees, what is the degree measure of angle AED? | <answer> 70° </answer> |
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In the parallelogram ABCD, if angle BAC measures 30 degrees and AC is equal to 2AB, what is the measure of angle D? then, please answer, As shown in the figure, in ABCD, angle ADC = 135° and angle CAD = 23°, then angle CAB is equal to (). | <answer> 22° </answer> |
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In the parallelogram ABCD, angle B measures 60 degrees and angle C measures 120 degrees. The line segment CE bisects angle ACB and intersects AD at point E. If the length of AE is 3 units, what is the length of CD? then, please answer, In the given diagram, quadrilateral ABCD is depicted with AE perpendicular to CD. If angle B measures 70 degrees, what is the degree measure of angle DAE? | <answer> 20° </answer> |
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In the parallelogram ABCD, if angle BAD measures 120 degrees and AC bisects angle BAD, what is the measure of angle BAC? then, please answer, In the given diagram depicting quadrilateral ABCD, if angle ABC measures 125 degrees and angle CAD measures 21 degrees, what is the measure of angle CAB? | <answer> 34° </answer> |
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In the parallelogram ABCD, point E is located on side AD and point F is located on side BC. The line segments BE and DF intersect at point G. If the area of triangle BGD is 12 square units and the area of triangle AEG is 6 square units, what is the area of triangle CEF? then, please answer, What is the measure of angle E in triangle BCDE if angle A in triangle ABC measures 40 degrees and AB is equal to AC? | <answer> 70° </answer> |
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In the parallelogram ABCD, if angle A measures 120 degrees and angle C measures 60 degrees, what is the measure of angle B? then, please answer, If angle A measures 60 degrees in parallelogram ABCD, what is the degree measure of angle C? | <answer> 60° </answer> |
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In the given figure, triangle ABC is depicted with points D and E serving as the midpoints of sides BC and AB respectively. Additionally, line segment DF is parallel to side AC and intersects side AB at point F. If the length of side AB measures 10 units, what is the length of side DF? then, please answer, In the given diagram, if in triangle ABC, point D is the midpoint of AC and point E is the midpoint of AB, and point F lies on the extension of BC, and if angle A measures 35° and angle AED measures 30°, then what is the degree measure of angle ACF? | <answer> 65° </answer> |
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In the parallelogram ABCD, if angle A measures 120 degrees and the bisector of angle ABC intersects CD at point E, what is the measure of angle CED? then, please answer, In the given diagram of quadrilateral ABCD, the angle bisector BE intersects DC at point E. If angle A measures 60°, what is the measure of angle DEB? | <answer> 120° </answer> |
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In the parallelogram ABCD, angle C is 120 degrees and angle BAD is 60 degrees. The bisector of angle BAD intersects CD at point E. If DE measures 3 units, what is the length of AB? then, please answer, As shown in the figure, in ABCD, BD = BC, AE is perpendicular to BD, and the foot is E. If angle C = 55°, then the degree of angle EAB is (). | <answer> 35° </answer> |
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In the parallelogram ABCD, if angle A measures 120 degrees and angle B is represented by x, what is the measure of angle C? then, please answer, What is the measurement of angle B in parallelogram ABCD if the sum of angles A and C is 130 degrees, as depicted in the diagram? | <answer> 115° </answer> |
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In the parallelogram ABCD, if angle B measures 120 degrees and angle C measures 60 degrees, what is the measure of angle D? then, please answer, If angle B in the parallelogram ABCD is 70 degrees, what is the measure of angle D? | <answer> 70° </answer> |
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In the parallelogram ABCD, AB is equal to 5 units and BC is equal to 3 units. The angle ACD measures 120 degrees. Determine the length of CD. then, please answer, In the given diagram, if the sum of angles A and C in parallelogram ABCD equals 160°, what is the measure of angle B? | <answer> 100° </answer> |
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In the parallelogram ABCD, if angle B measures 120 degrees and angle C measures 60 degrees, what is the degree measure of angle A? then, please answer, In the given parallelogram ABCD, if the sum of angle B and angle D measures 100°, what is the measure of angle A? | <answer> 130° </answer> |
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In the parallelogram ABCD, if angle B measures 120 degrees and angle C measures 60 degrees, what is the degree measure of angle A? then, please answer, In the given figure, if in quadrilateral ABCD, angle B added to angle D equals 140 degrees, what is the measure of angle C? | <answer> 110° </answer> |
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In the parallelogram ABCD, if angle ADB measures 30 degrees and angle CDB measures 120 degrees, what is the measure of angle CBD? then, please answer, As shown in the diagram, in the parallelogram ABCD, angle DBC = 25°, angle BAD = 115°, then angle BDC = (). | <answer> 40° </answer> |
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In the given figure, line l is parallel to line m. The angle bisectors of angles 1 and 2 intersect at point P. The angle bisector of angle 3 intersects line l at point Q. If angle 4 measures 50°, what is the measure of angle PQl? then, please answer, If the sum of angle 2, angle 3, and angle 4 is 320.0 degrees as indicated in the figure, what is the measure of angle 1? | <answer> 40度 </answer> |
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In the parallelogram ABCD, if angle B measures 120 degrees and AC bisects angle BAD, what is the measure of angle DAC? then, please answer, In the given parallelogram ABCD, if angle A measures 150 degrees, what is the degree measure of angle C? | <answer> 150° </answer> |
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In the parallelogram ABCD, if angle A measures 120 degrees and angle C measures 60 degrees, what is the measure of angle B? then, please answer, In the given diagram illustrating quadrilateral ABCD, if the sum of angle B and angle D is 200 degrees, what is the measure of angle A? | <answer> 80° </answer> |
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In the parallelogram ABCD, if angle ADB is equal to angle DBC and DE bisects angle ADC, intersecting BC at point E, and CE is equal to 2BE, what is the ratio of the area of triangle CDE to the area of triangle BCD? then, please answer, In the given diagram of quadrilateral ABCD, where AB equals BD and point E lies on BD with CE measuring the same as CB, if angle ADB is 65 degrees, what would be the measure of angle DCE? | <answer> 15° </answer> |
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In the parallelogram ABCD, AB is parallel to DM and BC is parallel to AM. The length of AD is 5 units and the length of AB is 3 units. If angle ADB measures 120 degrees, what is the length of BM? then, please answer, In the given diagram, for the quadrilateral ABCD, a point M is positioned on the extension of side AB. If angle D measures 120 degrees, what is the measure of angle CBM? | <answer> 60° </answer> |
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In the parallelogram ABCD, if angle A measures 120 degrees and angle B is represented by x, what is the measure of angle C? then, please answer, In the given figure, if the sum of angles A and C in quadrilateral ABCD is 110 degrees, what is the measure of angle B? | <answer> 125° </answer> |
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In the parallelogram ABCD, if angle BAC measures 30 degrees and AC bisects angle BAD, what is the measure of angle DCE? then, please answer, What is the measure of angle B in quadrilateral ABCD if point E lies on the extension of side DC, AE equals DE, and angle E measures 46 degrees? | <answer> 67° </answer> |
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In the parallelogram ABCD, if angle B measures 120 degrees and angle C measures 60 degrees, what is the measure of angle A? then, please answer, In the given figure for quadrilateral ABCD, if angle A measures twice the size of angle B, what is the measure of angle D? | <answer> 60° </answer> |
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In the parallelogram ABCD, if angle A measures 120 degrees and angle B is represented by x, what is the measure of angle C? then, please answer, What is the measure of angle D in parallelogram ABCD if the sum of angle A and angle C is 120 degrees, as depicted in the diagram? | <answer> 120° </answer> |
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In the parallelogram ABCD, if angle 1 measures 50°, what is the degree measure of angle B? then, please answer, If one side of quadrilateral ABCD, BC, is extended to point E and angle A measures 110° in the given diagram, what is the measure of angle 1? | <answer> 70° </answer> |
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In the given figure, triangle ABC is depicted with angle ACB measuring 90 degrees and angle ABO being twice the measure of angle BOC. The line segment BO intersects the extension of side AC at point D. If OD is perpendicular to AB and OB equals 2, what is the length of CD? then, please answer, If angle AOC is measured at 130.0 degrees in the figure depicting a set of triangular plates placed on a desktop (with the right-angle vertices coinciding), what is the measure of angle BOD? | <answer> 50° </answer> |
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In the parallelogram ABCD, if AE bisects angle BAD and intersects BC at point E, and if AB is equal to 2BE, what is the ratio of the area of triangle ADE to the area of triangle CDE? then, please answer, In the parallelogram ABCD, point E lies on side BC, and the line segment AE is drawn. If AE bisects angle BAD and angle D measures 58 degrees, what is the measure of angle AEC? | <answer> 119° </answer> |
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In the parallelogram ABCD, point E lies on side AD and BE is parallel to AC. The line segment DE intersects BC at point D. If the area of triangle ABE is 12 square units, what is the area of triangle CDE? then, please answer, In the given diagram of triangle ABC where AB is equal to AC, point D lies on side AC. By constructing quadrilateral BCDE with CB and CD as sides, if angle A measures 40°, what is the degree measure of angle E? | <answer> 70° </answer> |
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In the given diagram, quadrilateral ABCD is depicted with side AB parallel to side CD and side BC parallel to side AD. Point C lies on line segment BE, and angle 1 measures 50°. Determine the measure of angle ACD. then, please answer, As shown in the diagram, extend one side of ABCD, BC, to point E. If angle 1 = 80°, then angle A is equal to () | <answer> 100° </answer> |
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In the parallelogram ABCD, the line segment EF is parallel to AD and intersects BC at point E. The line segment DH is perpendicular to EF at point H. If the area of triangle DHB is 12 square units and the area of triangle DEF is 6 square units, what is the area of triangle AHD? then, please answer, In the given diagram, ABCD forms a parallelogram. At point E, DE is perpendicular to BC, and at point F, BF is perpendicular to CD. If DE and BF intersect at point H, and angle A measures 60°, then what is the measure of angle EHF? | <answer> 120° </answer> |
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In the parallelogram ABCD, if angle B measures 120 degrees and CE bisects angle DCE, what is the measure of angle BCE? then, please answer, In the given diagram, the quadrilateral ABCD is depicted. If the measure of angle A is 100 degrees and BC is extended to point E, what is the measure of angle DCE? | <answer> 80° </answer> |
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In the given diagram, triangle ABC is depicted with points D and E lying on the sides AB and BC respectively. The line segment DE is parallel to AC. Additionally, there is a line segment EF that intersects side AC at point F. If the ratio of BE to EC is 2:3 and the area of triangle DEF is 10 square units, what is the area of triangle ABC? then, please answer, If triangle ABC has angle B measuring 70 degrees and angle C measuring 50 degrees, and points D, E, and F are the midpoints of sides AB, BC, and CA respectively, what is the measure of angle DEF? | <answer> 60° </answer> |
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In the parallelogram ABCD, if angle A measures 120 degrees and angle B is represented by x, what is the measure of angle C? then, please answer, As shown in the figure, in ABCD, angle A+angle C=140°, then the measure of angle B is (). | <answer> 110° </answer> |
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In the given diagram, triangle ABC is depicted with points D and E serving as the midpoints of sides AB and BC respectively. The line segment DE intersects side AC at point C'. If the area of triangle ACD is 12 square units, what is the area of triangle BCD? then, please answer, In the provided figure, the perpendicular bisector of BC intersects AB and BC at point D and E, respectively. It also connects CD. If AC measures the same as DC and angle B is 25.0 degrees, what is the degree measure of angle ACD? | <answer> 80° </answer> |
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In the diagram, there is a triangle ABC where angle A measures 30 degrees and angle C measures 45 degrees. The line AB intersects the extension of side AC at point B, and the line BC intersects the extension of side AC at point C. The line AB is parallel to the line BC. Point D lies on the extension of side AB. The line AD intersects the extension of side BC at point E. The line DE intersects the extension of side AC at point F. The line EF intersects the extension of side AB at point G. The line FG intersects the extension of side AC at point H. The line GH then, please answer, In the given triangle ABC, if angle A measures 30 degrees, what is the sum of angle 1 and angle 2? | <answer> 210° </answer> |
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In the given figure, there is an equilateral triangle ABC with side length 10. A line segment DE is drawn parallel to BC and intersects AB at point D and AC at point E. If the area of triangle ADE is 25, what is the area of triangle ABC? then, please answer, What is the measure of angle α when a set of protractors are stacked together as shown in the diagram? | <answer> 75° </answer> |
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In the given diagram, triangle ABC is depicted with points D and E serving as the midpoints of sides AB and BC respectively. Additionally, line segment DE intersects side AC at point F. If the area of triangle ADF is 10 square units, what is the area of triangle CEF? then, please answer, As shown in the figure, if angle A = 20°, angle B = 45°, angle C = 35°, then angle DFE is equal to (). | <answer> 100° </answer> |
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In the given figure, triangle ABC is depicted with points D and E serving as the midpoints of sides AB and AC respectively. The line segment DE intersects side BC at point F. Additionally, there is a line passing through point A that intersects side BC at point G. If angle 1 measures 30 degrees and angle 2 measures 45 degrees, what is the measure of angle B? then, please answer, In the given diagram, if triangle ABC is folded along DE and point A ends up outside quadrilateral BCDE, with angle 1 measuring 70° and angle 2 measuring 152°, what is the measure of angle A? | <answer> 42° </answer> |
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In the given figure, triangle ABC is depicted with points D and E located on sides AB and AC respectively. The line segment DE is parallel to side BC. If the ratio of AD to DB is 2:3 and the area of triangle ABD is 10 square units, what is the area of triangle CDE? then, please answer, In the given diagram, BD is the angle bisector of triangle ABC. If angle DBA is 30 degrees and angle ADB is 80 degrees, what is the measure of angle C? | <answer> 50° </answer> |
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In the given figure, triangle ABC is depicted with points D and E located on sides AB and AC respectively. The line segment DE is parallel to side BC. Additionally, there is a point F on side BC such that DF is perpendicular to EF. If the area of triangle AEF is 12 square units and the area of triangle DEF is 3 square units, what is the area of triangle ABC? then, please answer, What is the sum of the measures of angles A, B, C, D, E, and F in the given diagram? | <answer> 360° </answer> |
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In the given diagram, triangle ABC is depicted with angle ACB measuring 90 degrees and angle BAC measuring 30 degrees. Point P lies on the extension of side BC. The bisector of angle APB intersects side AC at point A', and the bisector of angle PCA intersects side AB at point B'. If the length of segment PA' is 2, what is the length of segment PB'? then, please answer, BP is the bisector of angle ABC, CP is the bisector of the adjacent supplementary angle of angle ACB. angle ABP = 20°, angle ACP = 50°, then angle P = ( ). | <answer> 30° </answer> |
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The diagram shows a parallelogram ABCD, where AB is parallel to CD and AD is parallel to BC. The angle bisector of angle BAD intersects the extension of side BC at point E. If the length of AE is 10 units and BE is 6 units, what is the perimeter of parallelogram ABCD? then, please answer, What is the measure of angle 2 if a triangle ruler with a 30.0 angle is placed on the ruler and angle 1 measures 25.0 degrees in the given figure? | <answer> 35° </answer> |
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In the given figure, there is a quadrilateral ABCD where AB is parallel to CD and AD is parallel to BC. The angle A measures 100 degrees, while angle B measures 80 degrees. Additionally, the length of side AB is 6 units. Determine the perimeter of the quadrilateral ABCD. then, please answer, What method can be used to determine the measurement of angle C in triangle ABC before any damage occurs? | <answer> 65° </answer> |
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In the given diagram, triangle ABC is depicted with points D and E lying on the extensions of sides AB and AC respectively. The line DE intersects BC at point H. Additionally, there is a line passing through point C that intersects the extension of side EF at point G. If angle A measures 40 degrees and angle F measures 35 degrees, what is the measure of angle G? then, please answer, As shown in the diagram, given angle A = 60°, the sum of the degrees of angle D + angle E + angle F + angle G is () | <answer> 240° </answer> |
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In the given figure, there is a pentagon ABCDE with vertices A, B, C, D, and E. The midpoints of sides AB, BC, CD, DE, and EA are labeled as F, G, H, I, and J respectively. Additionally, the midpoints of segments FG, GH, HI, and IJ are labeled as K, L, M, and N respectively. If the area of triangle PAB is 12 square units, what is the area of quadrilateral PFGN? then, please answer, In the given heptagon ABCDEFG, the lines EF and BA are extended to intersect at point P. If the total of the exterior angles of angles ABC, BCD, CDE, and DEF is 230°, what is the measurement of angle P? | <answer> 50° </answer> |
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In the given figure, triangle ABC is depicted with points D and E located on sides AB and AC respectively. The line segment DE is parallel to BC. If AD measures 3 units, DB measures 2 units, and AE measures 6 units, what is the length of EC? then, please answer, In the provided diagram, if angle A measures 70°, angle B measures 40°, and angle C measures 30°, what is the total measure of angles D and E? | <answer> 40° </answer> |
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In the given figure, triangle ABC is depicted with points C and D lying on line BD. The angle ACB measures 40 degrees, and the angle ACD measures 120 degrees. then, please answer, In the given diagram, points B, C, and D are aligned in a straight line. If angle B measures 56° and angle ACD measures 120°, what is the measure of angle A? | <answer> 64° </answer> |
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In the given figure, triangle ABC is depicted with points O and D located on the line segments AB and BC respectively. The line segment OD intersects the extension of AC at point E. If angle ABO measures 20 degrees and angle CBE measures 30 degrees, what is the measure of angle COE? then, please answer, Triangle ABC's interior angle bisectors OB and OC intersect at point O. If angle A = 110°, then angle BOC = () | <answer> 145° </answer> |
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In the given diagram, triangle ABC is depicted with angle ACB measuring 90 degrees and angle CAB measuring 30 degrees. A circle with center C and radius CA intersects side AB at point D. The line segment CD intersects side BC at point E. If angle AEB is represented by α, what is the value of α? then, please answer, What is the measure of angle α in the triangle board puzzle activity stacked by Xiaoming according to the given diagram? | <answer> 105° </answer> |
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In the given diagram, circle O has diameter AB and point C lies on the circumference of the circle. The line OC intersects the tangent line BA at point A and the extension of the chord BC at point C. If angle B measures 30 degrees, what is the measure of angle BAC? then, please answer, In the given diagram, circle O has chord AB and tangent AC with point A as the tangent point. Additionally, line BC passes through the center of the circle. If angle B measures 25.0, what is the measure of angle C? | <answer> 40° </answer> |
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In the given diagram, there is a triangle ABC where AB and AC are equal in length. The vertices A, B, and C are located on the grid points. then, please answer, In the provided grid diagram, the vertices of triangle ABC are situated on the grid points, and the side length of each small square is 1. What is the measure of angle ABC? | <answer> 135° </answer> |
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In the circle O, AB is the diameter and CD is a chord. The perpendicular bisector of CD intersects AB at point E. If angle AOC measures 100°, what is the measure of angle B? then, please answer, In the given diagram, if BD is the diameter of circle O, and BD is perpendicular to AC at point E, with angle AOC measuring 100 degrees, what is the measure of angle OCD? | <answer> 25° </answer> |
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In the given figure, circle O has diameter AB and chord BC. Point C is located on the circumference of circle O. The line OC intersects the extension of chord AC at point D. If angle BOC measures 70°, what is the measure of angle ADC? then, please answer, If BC is the diameter of circle O and angle AOC measures 58 degrees, what is the measure of angle A? | <answer> 29° </answer> |
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In the given figure, triangle ABC is depicted with points O and D located on its sides AB and AC respectively. The line segment OD intersects side BC at point E. Additionally, there is a circle that passes through points A, B, and C. This circle intersects side BC at point F. If angle BOD measures 100°, what is the measure of angle C? then, please answer, As shown in the figure, triangle ABC is circumscribed to circle O and angle A is 40°. Therefore, the degree of angle BCO is () | <answer> 50° </answer> |
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In the given diagram, circle O has diameter AB and chord AC. Point C lies on the circumference of circle O. The line segment OC intersects the extension of chord AB at point D. If angle AOC measures 100°, what is the measure of angle ADC? then, please answer, In the given diagram, if angle C in circle O measures 20 degrees and angle B is 35 degrees, what is the measure of angle A? | <answer> 15° </answer> |
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In the given diagram, circle O is depicted with AB serving as its diameter. Points C and D lie on the circumference of circle O, while points E and A are also on the circumference. The line segment DE intersects circle O at point B. Additionally, there is a line segment CE that is parallel to AB. then, please answer, In the given diagram, if AB is the diameter of circle O and points C, D, and E are located on circle O, with AD equal to CD and angle E measuring 70°, what is the measure of angle ABC? | <answer> 40° </answer> |
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In the given diagram, circle O is depicted with points A and B lying on its circumference. The line segment AB serves as the diameter of circle O. Point C is positioned outside circle O. Connecting AC and BC forms triangle ABC. Additionally, OC intersects AB at point D. If angle CAB measures 30 degrees, what is the measure of angle ADB? then, please answer, In the given diagram, if triangle ABC is inscribed in circle O and the measure of angle ABO is 40 degrees, what is the measure of angle ACB? | <answer> 50° </answer> |
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In the given figure, circle O is tangent to line BC at point B and intersects lines AB and AC at points D and E respectively. The extension of line DE intersects circle O at point F. If angle A measures 50°, what is the measure of angle C? then, please answer, If quadrilateral ABCD is circumscribed around circle O and one of its exterior angles, angle DCE, measures 115°, then what is the measure of the other exterior angle, angle DAF? | <answer> 65° </answer> |
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In the given diagram, circle O is depicted with diameter AB and chord CD. Point P lies on the extension of AB beyond point B. A line segment CP intersects circle O at point C and another line segment PD intersects circle O at point D. The line segment AD is perpendicular to line segment BP. If angle ADC measures 30 degrees, what is the measure of angle B? then, please answer, In the given diagram, all the points A, B, C, D, and P are located on the circle O. The line OC is perpendicular to AB. When angle ADC measures 15° (where 0° < α < 90°), what is the measure of angle APB? | <answer> 150° </answer> |
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In the given diagram, there is a cone with a base radius of 3 and a slant height of 5. The axis of the cone intersects the base at point O, which is also the center of the base. A line segment AB is drawn parallel to the slant height, intersecting the lateral surface of the cone at points A and B. If the angle between the line segment OA and the slant height is 60°, what is the length of line segment AB? then, please answer, What is the lateral area of the conical funnel made by Xiao Ming in the comprehensive practical activity class if the bottom radius is 6.0 and the height is 8.0? | <answer> 60πcm </answer> |
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In the given figure, triangle ABC is depicted with points D and E located on the circumcircle of triangle ABC. The line DE intersects side AB at point O. If angle AOB measures 120 degrees, what is the measure of angle C? then, please answer, In the given diagram, AB is the diameter of circle O, and points C and D lie on circle O. If angle D measures 120°, what is the measure of angle CAB? | <answer> 30° </answer> |
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In the given diagram, circle O has diameter AD and chord CE. The line OE is tangent to circle O at point E. Point D lies on the extension of line AE. If angle AOD measures 70°, what is the measure of angle C? then, please answer, If CD is the diameter of circle O and chord DE is parallel to radius OA, while arc CE measures 92 degrees, what is the measure of angle C? | <answer> 23° </answer> |
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In the given diagram, circle O is depicted with points A and C lying on its circumference. The line segment AB connects points A and B, while OC intersects AB at point O. Additionally, line segment AC bisects angle BOC. then, please answer, If point A, B, and C are on circle O and angle B measures 32 degrees, what is the measure of angle OAC? | <answer> 58° </answer> |
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In the given diagram, circle O has diameter AB. Points C and D lie on the circumference of circle O. The line segment CD is perpendicular to the diameter AB at point B. Additionally, angle AOC measures 100 degrees. Determine the measure of angle ADB. then, please answer, In the provided diagram, points A, B, and C lie on the circle O in the given order. If the difference between angle B and angle A is 40 degrees, what is the measure of angle AOB? | <answer> 80° </answer> |
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In the given figure, quadrilateral ABCD is inscribed within circle O. The extension of side AD intersects circle O at point E. If angle A measures 100° and angle B measures 70°, what is the measure of angle CED? then, please answer, Given the figure with points A, B, C, D, and E all lying on circle O, and AC equal to AE, if angle B measures 128 degrees, what is the degree measure of angle D? | <answer> 104° </answer> |
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In the given diagram, circle O is depicted with points A, B, and C lying on its circumference. The line segment AB serves as the diameter of circle O. Additionally, point O is connected to both points A and C. then, please answer, As shown in the diagram, point A, B, and C are three points on circle O. If angle AOB = 66°, then the degree of angle C is () | <answer> 33° </answer> |
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In the given diagram, circle O is depicted with points A, B, C, and D lying on its circumference. The line segment AB serves as the diameter of circle O. Additionally, there is a line segment CD that intersects circle O at point E. Furthermore, angle ACD measures 30 degrees. then, please answer, In the given diagram, if quadrilateral ABCD is inscribed in circle O and angle BCD measures 140°, what is the degree measure of angle BOD? | <answer> 80° </answer> |
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In the given diagram, circle O has diameter OC and point A lies on the circumference of circle O. The line segment OA is extended to intersect circle O at point B. Point C is located on the circumference of circle O. If angle AOC measures 100°, what is the measure of angle ABC? then, please answer, Given that points A, B, and C all lie on circle O, and angle BOC measures 100 degrees, what is the measure of angle BAC? | <answer> 50° </answer> |
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In the given figure, circle O has diameter AB and chord CD is perpendicular to AB at point O. The extended line of AO intersects circle O at point D. If angle BAC measures 30 degrees, what is the measure of angle ACD? then, please answer, As shown in the figure, AB is the diameter of circle O. C and D are two points on circle O. Connect AC, CD, and AD. If angle ADC = 75°, then the degree of angle BAC is (). | <answer> 15° </answer> |
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In the given diagram, circle O has diameter AB and chord CD. The perpendicular bisector of chord CD intersects AB at point O and intersects circle O at point D. If angle AOB measures 70°, what is the measure of angle C? then, please answer, In the given diagram, circle O has line AB as its diameter. Point D lies on circle O, and OD and BD are connected. A tangent is drawn to circle O passing through point D, and it intersects the extension of line BA at point C. If angle C measures 40°, what is the measure of angle B? | <answer> 25° </answer> |
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In the given diagram, there is a circle O with diameter AB. Point C lies on the circumference of circle O and is connected to points A and B. The line OC intersects the tangent line AB at point D. If angle AOB measures 60°, what is the measure of angle OCD? then, please answer, ? | <answer> 100π平方米 </answer> |
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In the given diagram, circle O is depicted with diameter AB. Point C lies on the circumference of circle O and is connected to points A and B. The line segment CD is tangent to circle O at point C. Line segment DE intersects circle O at point D and extends beyond point C. Additionally, line segment BE is parallel to line segment AC. If angle E measures 20 degrees, what is the measure of angle A? then, please answer, As shown in the figure, quadrilateral ABCD is inscribed in circle O, with DA=DC. If angle CBE=45°, then the measure of angle DAC is () | <answer> 67.5° </answer> |
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In the given diagram, circle O is depicted with points A, B, and C lying on its circumference. The line segment AB serves as the diameter of circle O. Additionally, angle ABC measures 30 degrees. then, please answer, What is the measure of angle AOB if angle ACB measures 36 degrees and points A, B, and C are all located on circle O? | <answer> 72° </answer> |
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In the given diagram, circle O is depicted with diameter AB and chord CD. The line OE is perpendicular to CD at point E. Additionally, the line AD intersects the extension of OE at point F. If angle AOB measures 100°, what is the measure of angle C? then, please answer, If in the given figure, circle O is the circumcircle of triangle ABC, and angle A measures 62°, and E is the midpoint of BC, what is the degree measure of angle D where OE, extended, intersects circle O at point D, and BD is then connected? | <answer> 59° </answer> |
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In the given diagram, triangle ABC is inscribed within circle O. The center of the circle is denoted as point O. The radius of circle O is 5 units. The measure of angle BAC is 60 degrees. Determine the length of side BC. then, please answer, In the given diagram of circle O, if the length of AB is equal to the length of AC and angle B measures 70 degrees, what is the measure of angle A? | <answer> 40° </answer> |
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In the given figure, circle O is depicted with AB as its diameter and CD as a chord. The tangent line DE intersects circle O at point E. If angle BAC measures 30°, what is the measure of angle AED? then, please answer, The degree of angle ADE is (). | <answer> 100° </answer> |
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