id int32 1 6.98k | annotation stringclasses 132
values | source stringlengths 7 17 | problem_level int32 0 28 | problem_text_cn stringlengths 20 201 | problem_text_en stringlengths 58 424 | problem_img stringlengths 5 8 | construction_cdl listlengths 1 28 | text_cdl listlengths 0 16 | image_cdl listlengths 0 16 | goal_cdl stringlengths 8 131 | problem_answer stringclasses 906
values | theorem_seqs listlengths 0 28 | theorem_seqs_dag_json stringlengths 13 3.3k | image imagewidth (px) 48 1.6k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6,801 | ChenyangMao_2023-06-04 | GeoQA-angle-10007 | 7 | 如图所示，BA=BC，AD平分∠CAE，DE是AB的垂直平分线。求∠EBD的大小。 | As shown in the diagram, BA=BC, AD is the angle bisector of ∠CAE, DE perpendicular bisects AB. Find the measure of ∠EBD. | 6801.png | [
"Shape(BD,DE,EB)",
"Shape(AE,ED,DA)",
"Shape(AD,DC,CA)",
"Collinear(BDC)",
"Collinear(AEB)"
] | [
"Equal(LengthOfLine(BA),LengthOfLine(BC))",
"IsBisectorOfAngle(AD,CAE)",
"IsPerpendicularBisectorOfLine(DE,AB)"
] | [] | Value(MeasureOfAngle(EBD)) | 36 | [
"perpendicular_bisector_property_distance_equal(1,DE,AB)",
"angle_addition(1,CAD,DAE)",
"triangle_property_angle_sum(1,BCA)",
"isosceles_triangle_judgment_line_equal(1,DAB)",
"isosceles_triangle_judgment_line_equal(1,BCA)",
"isosceles_triangle_property_angle_equal(1,DAB)",
"isosceles_triangle_property_a... | {"START": ["perpendicular_bisector_property_distance_equal(1,DE,AB)", "angle_addition(1,CAD,DAE)", "triangle_property_angle_sum(1,BCA)", "isosceles_triangle_judgment_line_equal(1,BCA)"], "isosceles_triangle_judgment_line_equal(1,BCA)": ["isosceles_triangle_property_angle_equal(1,BCA)"], "isosceles_triangle_judgment_lin... | |
6,802 | ChenyangMao_2023-06-04 | GeoQA-angle-10008 | 7 | 如图所示，∠ACD=20°，B是⊙B的圆心，三角形ACB为直角三角形。求∠BAC的大小。 | As shown in the diagram, ∠ACD=20°, the center of circle B is B, AC and CB are the legs of the right △ ACB. Find the measure of ∠BAC. | 6802.png | [
"Shape(DA,AC,BDC)",
"Shape(BDC,CD)",
"Shape(BD,DC,CB)",
"Collinear(ADB)",
"Cocircular(B,DC)"
] | [
"Equal(MeasureOfAngle(ACD),20)",
"IsCentreOfCircle(B,B)",
"RightTriangle(ACB)"
] | [] | Value(MeasureOfAngle(BAC)) | 50 | [
"angle_addition(1,ACD,DCB)",
"triangle_property_angle_sum(1,BDC)",
"triangle_property_angle_sum(1,ACB)",
"radius_of_circle_property_length_equal(1,BD,B)",
"radius_of_circle_property_length_equal(1,BC,B)",
"isosceles_triangle_judgment_line_equal(1,BDC)",
"isosceles_triangle_property_angle_equal(1,BDC)"
] | {"START": ["angle_addition(1,ACD,DCB)", "triangle_property_angle_sum(1,BDC)", "triangle_property_angle_sum(1,ACB)", "radius_of_circle_property_length_equal(1,BD,B)", "radius_of_circle_property_length_equal(1,BC,B)"], "isosceles_triangle_judgment_line_equal(1,BDC)": ["isosceles_triangle_property_angle_equal(1,BDC)"], "r... | |
6,803 | ChenyangMao_2023-06-04 | GeoQA-angle-10009 | 6 | 如图所示，AB=AC，∠ABC=25°，AD是△ABC的中线。求∠DAB的大小。 | As shown in the diagram, AB=AC, ∠ABC=25°, AD is the median of triangle ABC. Find the measure of ∠DAB. | 6803.png | [
"Shape(AB,BD,DA)",
"Shape(AD,DC,CA)",
"Collinear(BDC)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(AC))",
"Equal(MeasureOfAngle(ABC),25)",
"IsMedianOfTriangle(AD,ABC)"
] | [] | Value(MeasureOfAngle(DAB)) | 65 | [
"isosceles_triangle_judgment_line_equal(1,ABC)",
"isosceles_triangle_property_angle_equal(1,ABC)",
"triangle_property_angle_sum(1,ABD)",
"mirror_congruent_triangle_judgment_sas(1,BDA,CAD)",
"mirror_congruent_triangle_property_angle_equal(1,DAB,DCA)",
"adjacent_complementary_angle(1,BDA,ADC)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,ABC)", "triangle_property_angle_sum(1,ABD)", "adjacent_complementary_angle(1,BDA,ADC)"], "isosceles_triangle_judgment_line_equal(1,ABC)": ["isosceles_triangle_property_angle_equal(1,ABC)"], "isosceles_triangle_property_angle_equal(1,ABC)": ["mirror_congruent_triangle... | |
6,804 | ChenyangMao_2023-06-04 | GeoQA-angle-10011 | 5 | 如图所示，OC=OE，∠BAO=50°，AB平行于OD。求∠OCE的大小。 | As shown in the diagram, OC=OE, ∠BAO=50°, AB is parallel to OD. Find the measure of ∠OCE. | 6804.png | [
"Shape(BA,AO)",
"Shape(AO,OD)",
"Shape(DO,OE)",
"Shape(OC,CE,EO)",
"Collinear(AOE)",
"Collinear(COD)"
] | [
"Equal(LengthOfLine(OC),LengthOfLine(OE))",
"Equal(MeasureOfAngle(BAO),50)",
"ParallelBetweenLine(AB,OD)"
] | [] | Value(MeasureOfAngle(OCE)) | 25 | [
"parallel_property_corresponding_angle(2,AB,OD,E)",
"adjacent_complementary_angle(1,DOE,EOC)",
"triangle_property_angle_sum(1,OCE)",
"isosceles_triangle_judgment_line_equal(1,OCE)",
"isosceles_triangle_property_angle_equal(1,OCE)"
] | {"START": ["parallel_property_corresponding_angle(2,AB,OD,E)", "adjacent_complementary_angle(1,DOE,EOC)", "triangle_property_angle_sum(1,OCE)", "isosceles_triangle_judgment_line_equal(1,OCE)"], "isosceles_triangle_judgment_line_equal(1,OCE)": ["isosceles_triangle_property_angle_equal(1,OCE)"]} | |
6,805 | ChenyangMao_2023-06-04 | GeoQA-angle-10012 | 4 | 如图所示，∠DEA=40°，三角形ABC为等边三角形，DE∥PC，PC∥BF。求∠CBF的大小。 | As shown in the diagram, ∠DEA=40°, triangle ABC is an equilateral triangle, DE∥PC, PC is parallel to BF. Find the measure of ∠CBF. | 6805.png | [
"Shape(AD,DE,EA)",
"Shape(DP,PC,CE,ED)",
"Shape(PB,BC,CP)",
"Shape(CB,BF)",
"Collinear(ADPB)",
"Collinear(AEC)"
] | [
"Equal(MeasureOfAngle(DEA),40)",
"EquilateralTriangle(ABC)",
"ParallelBetweenLine(DE,PC)",
"ParallelBetweenLine(PC,BF)"
] | [] | Value(MeasureOfAngle(CBF)) | 20 | [
"equilateral_triangle_property_angle(1,CAB)",
"parallel_property_corresponding_angle(2,CP,ED,A)",
"angle_addition(1,BCP,PCA)",
"parallel_property_alternate_interior_angle(2,PC,BF)"
] | {"START": ["equilateral_triangle_property_angle(1,CAB)", "parallel_property_corresponding_angle(2,CP,ED,A)", "angle_addition(1,BCP,PCA)", "parallel_property_alternate_interior_angle(2,PC,BF)"]} | |
6,806 | ChenyangMao_2023-06-04 | GeoQA-angle-10013 | 8 | 如图所示，AD=AO，三角形ABC为等边三角形，BO是三角形BCA的中线。求∠BOD的大小。 | As shown in the diagram, AD=AO, triangle ABC is an equilateral triangle, BO is the median of △ BCA. Find the measure of ∠BOD. | 6806.png | [
"Shape(OD,DA,AO)",
"Shape(CO,OB,BC)",
"Shape(OA,AB,BO)",
"Collinear(DAB)",
"Collinear(COA)"
] | [
"Equal(LengthOfLine(AD),LengthOfLine(AO))",
"EquilateralTriangle(ABC)",
"IsMedianOfTriangle(BO,BCA)"
] | [] | Value(MeasureOfAngle(BOD)) | 120 | [
"equilateral_triangle_property_angle(1,ABC)",
"adjacent_complementary_angle(1,DAO,OAB)",
"isosceles_triangle_judgment_line_equal(1,AOD)",
"isosceles_triangle_property_angle_equal(1,AOD)",
"triangle_property_angle_sum(1,ODA)",
"triangle_property_angle_sum(1,OAB)",
"triangle_property_angle_sum(1,ODB)",
... | {"START": ["equilateral_triangle_property_angle(1,ABC)", "adjacent_complementary_angle(1,DAO,OAB)", "isosceles_triangle_judgment_line_equal(1,AOD)", "triangle_property_angle_sum(1,ODA)", "triangle_property_angle_sum(1,OAB)", "triangle_property_angle_sum(1,ODB)", "isosceles_triangle_property_line_coincidence(2,BCA,O)"],... | |
6,807 | ChenyangMao_2023-06-04 | GeoQA-angle-10014 | 4 | 如图所示，AC=BC，∠ECD=70°，CD平行于AB。求∠BCA的大小。 | As shown in the diagram, AC=BC, ∠ECD=70°, CD∥AB. Find the measure of ∠BCA. | 6807.png | [
"Shape(EC,CD)",
"Shape(DC,CB)",
"Shape(CA,AB,BC)",
"Collinear(ECA)"
] | [
"Equal(LengthOfLine(AC),LengthOfLine(BC))",
"Equal(MeasureOfAngle(ECD),70)",
"ParallelBetweenLine(CD,AB)"
] | [] | Value(MeasureOfAngle(BCA)) | 40 | [
"isosceles_triangle_judgment_line_equal(1,CAB)",
"isosceles_triangle_property_angle_equal(1,CAB)",
"triangle_property_angle_sum(1,CAB)",
"parallel_property_corresponding_angle(1,CD,AB,E)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,CAB)", "triangle_property_angle_sum(1,CAB)", "parallel_property_corresponding_angle(1,CD,AB,E)"], "isosceles_triangle_judgment_line_equal(1,CAB)": ["isosceles_triangle_property_angle_equal(1,CAB)"]} | |
6,808 | ChenyangMao_2023-06-04 | GeoQA-angle-10020 | 7 | 如图所示，AD=BC，∠EPF=130°，E平分线段AB，F平分线段CD，P是线段DB的中点。求∠FEP的大小。 | As shown in the diagram, AD=BC, ∠EPF=130°, E is the midpoint of segment AB, F is the midpoint of segment CD, P bisects segment DB. Find the measure of ∠FEP. | 6808.png | [
"Shape(DA,AE,EP,PD)",
"Shape(PE,EB,BP)",
"Shape(PF,FD,DP)",
"Shape(CF,FP,PB,BC)",
"Shape(FE,EP,PF)",
"Collinear(CFD)",
"Collinear(DPB)",
"Collinear(AEB)"
] | [
"Equal(LengthOfLine(AD),LengthOfLine(BC))",
"Equal(MeasureOfAngle(EPF),130)",
"IsMidpointOfLine(E,AB)",
"IsMidpointOfLine(F,CD)",
"IsMidpointOfLine(P,DB)"
] | [] | Value(MeasureOfAngle(FEP)) | 25 | [
"midsegment_of_triangle_judgment_midpoint(1,PE,BDA)",
"midsegment_of_triangle_judgment_midpoint(1,PF,DBC)",
"midsegment_of_triangle_property_length(1,PE,BDA)",
"midsegment_of_triangle_property_length(1,PF,DBC)",
"isosceles_triangle_judgment_line_equal(1,PFE)",
"triangle_property_angle_sum(1,FEP)",
"isos... | {"START": ["midsegment_of_triangle_judgment_midpoint(1,PE,BDA)", "midsegment_of_triangle_judgment_midpoint(1,PF,DBC)", "triangle_property_angle_sum(1,FEP)"], "isosceles_triangle_judgment_line_equal(1,PFE)": ["isosceles_triangle_property_angle_equal(1,PFE)"], "midsegment_of_triangle_judgment_midpoint(1,PE,BDA)": ["midse... | |
6,809 | ChenyangMao_2023-06-04 | GeoQA-angle-10023 | 4 | 如图所示，∠ABC=50°，∠CAB=60°，D是线段AB的中点，E是线段AC的中点。求∠DEA的大小。 | As shown in the diagram, ∠ABC=50°, ∠CAB=60°, D bisects segment AB, E is the midpoint of segment AC. Find the measure of ∠DEA. | 6809.png | [
"Shape(AD,DE,EA)",
"Shape(DB,BC,CE,ED)",
"Collinear(ADB)",
"Collinear(AEC)"
] | [
"Equal(MeasureOfAngle(ABC),50)",
"Equal(MeasureOfAngle(CAB),60)",
"IsMidpointOfLine(D,AB)",
"IsMidpointOfLine(E,AC)"
] | [] | Value(MeasureOfAngle(DEA)) | 70 | [
"triangle_property_angle_sum(1,ABC)",
"midsegment_of_triangle_judgment_midpoint(1,DE,ABC)",
"midsegment_of_triangle_property_parallel(1,DE,ABC)",
"parallel_property_corresponding_angle(2,CB,ED,A)"
] | {"START": ["triangle_property_angle_sum(1,ABC)", "midsegment_of_triangle_judgment_midpoint(1,DE,ABC)"], "midsegment_of_triangle_judgment_midpoint(1,DE,ABC)": ["midsegment_of_triangle_property_parallel(1,DE,ABC)"], "midsegment_of_triangle_property_parallel(1,DE,ABC)": ["parallel_property_corresponding_angle(2,CB,ED,A)"]... | |
6,810 | ChenyangMao_2023-06-04 | GeoQA-angle-10024 | 2 | 如图所示，∠CDA=135°，∠DAC=23°，四边形ABCD是平行四边形。求∠CAB的大小。 | As shown in the diagram, ∠CDA=135°, ∠DAC=23°, BA and CD are opposite sides of the ▱ ABCD. Find the measure of ∠CAB. | 6810.png | [
"Shape(CD,DA,AC)",
"Shape(CA,AB,BC)"
] | [
"Equal(MeasureOfAngle(CDA),135)",
"Equal(MeasureOfAngle(DAC),23)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(CAB)) | 22 | [
"parallel_property_ipsilateral_internal_angle(1,DC,AB)",
"angle_addition(1,DAC,CAB)"
] | {"START": ["parallel_property_ipsilateral_internal_angle(1,DC,AB)", "angle_addition(1,DAC,CAB)"]} | |
6,811 | ChenyangMao_2023-06-04 | GeoQA-angle-10025 | 2 | 如图所示，∠ABC=70°，四边形ABCD是平行四边形，AE⊥DE。求∠DAE的大小。 | As shown in the diagram, ∠ABC=70°, BA and CD are opposite sides of the parallelogram ABCD, AE is perpendicular to DE. Find the measure of ∠DAE. | 6811.png | [
"Shape(AB,BC,CE,EA)",
"Shape(AE,ED,DA)",
"Collinear(CED)"
] | [
"Equal(MeasureOfAngle(ABC),70)",
"Parallelogram(ABCD)",
"PerpendicularBetweenLine(AE,DE)"
] | [] | Value(MeasureOfAngle(DAE)) | 20 | [
"parallelogram_property_opposite_angle_equal(1,BCDA)",
"triangle_property_angle_sum(1,AED)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,BCDA)", "triangle_property_angle_sum(1,AED)"]} | |
6,812 | ChenyangMao_2023-06-04 | GeoQA-angle-10027 | 2 | 如图所示，∠ABC=125°，∠DAC=21°，BA和CD是平行四边形ABCD的一组对边。求∠CAB的大小。 | As shown in the diagram, ∠ABC=125°, ∠DAC=21°, ABCD is a ▱. Find the measure of ∠CAB. | 6812.png | [
"Shape(AB,BC,CA)",
"Shape(AC,CD,DA)"
] | [
"Equal(MeasureOfAngle(ABC),125)",
"Equal(MeasureOfAngle(DAC),21)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(CAB)) | 34 | [
"parallel_property_ipsilateral_internal_angle(1,AD,BC)",
"angle_addition(1,DAC,CAB)"
] | {"START": ["parallel_property_ipsilateral_internal_angle(1,AD,BC)", "angle_addition(1,DAC,CAB)"]} | |
6,813 | ChenyangMao_2023-06-04 | GeoQA-angle-10028 | 4 | 如图所示，AB=AC，∠BAC=40°，EB和DC是▱BEDC的一组对边。求∠BED的大小。 | As shown in the diagram, AB=AC, ∠BAC=40°, BEDC is a ▱. Find the measure of ∠BED. | 6813.png | [
"Shape(AD,DC,CB,BA)",
"Shape(BE,ED,DC,CB)",
"Collinear(ADC)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(AC))",
"Equal(MeasureOfAngle(BAC),40)",
"Parallelogram(BEDC)"
] | [] | Value(MeasureOfAngle(BED)) | 70 | [
"isosceles_triangle_judgment_line_equal(1,ACB)",
"isosceles_triangle_property_angle_equal(1,ACB)",
"triangle_property_angle_sum(1,ACB)",
"parallelogram_property_opposite_angle_equal(1,EDCB)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,ACB)", "triangle_property_angle_sum(1,ACB)", "parallelogram_property_opposite_angle_equal(1,EDCB)"], "isosceles_triangle_judgment_line_equal(1,ACB)": ["isosceles_triangle_property_angle_equal(1,ACB)"]} | |
6,814 | ChenyangMao_2023-06-04 | GeoQA-angle-10029 | 1 | 如图所示，∠DAB=60°，ABCD是平行四边形。求∠BCD的大小。 | As shown in the diagram, ∠DAB=60°, BA and CD are opposite sides of the ▱ ABCD. Find the measure of ∠BCD. | 6814.png | [
"Shape(AB,BC,CD,DA)"
] | [
"Equal(MeasureOfAngle(DAB),60)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(BCD)) | 60 | [
"parallelogram_property_opposite_angle_equal(1,ABCD)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,ABCD)"]} | |
6,815 | ChenyangMao_2023-06-04 | GeoQA-angle-10030 | 8 | 如图所示，AE=BE，BC=BE，∠CDA=105°，四边形ABCD是▱。求∠CAB的大小。 | As shown in the diagram, AE=BE, BC=BE, ∠CDA=105°, quadrilateral ABCD is a ▱. Find the measure of ∠CAB. | 6815.png | [
"Shape(AB,BE,EA)",
"Shape(CE,EB,BC)",
"Shape(DA,AE,EC,CD)",
"Collinear(AEC)"
] | [
"Equal(LengthOfLine(AE),LengthOfLine(BE))",
"Equal(LengthOfLine(BC),LengthOfLine(BE))",
"Equal(MeasureOfAngle(CDA),105)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(CAB)) | 25 | [
"parallelogram_property_opposite_angle_equal(1,BCDA)",
"angle_addition(1,ABE,EBC)",
"isosceles_triangle_judgment_line_equal(1,EAB)",
"isosceles_triangle_judgment_line_equal(1,BCE)",
"isosceles_triangle_property_angle_equal(1,EAB)",
"isosceles_triangle_property_angle_equal(1,BCE)",
"triangle_property_ang... | {"START": ["parallelogram_property_opposite_angle_equal(1,BCDA)", "angle_addition(1,ABE,EBC)", "isosceles_triangle_judgment_line_equal(1,EAB)", "isosceles_triangle_judgment_line_equal(1,BCE)", "triangle_property_angle_sum(1,CEB)", "triangle_property_angle_sum(1,ABC)"], "isosceles_triangle_judgment_line_equal(1,BCE)": [... | |
6,816 | ChenyangMao_2023-06-04 | GeoQA-angle-10031 | 5 | 如图所示，∠AED=30°，∠CAB=35°，D平分线段AC，E平分线段AB。求∠ACF的大小。 | As shown in the diagram, ∠AED=30°, ∠CAB=35°, D bisects segment AC, E is the midpoint of segment AB. Find the measure of ∠ACF. | 6816.png | [
"Shape(AE,ED,DA)",
"Shape(EB,BC,CD,DE)",
"Shape(AC,CF)",
"Collinear(AEB)",
"Collinear(ADC)",
"Collinear(BCF)"
] | [
"Equal(MeasureOfAngle(AED),30)",
"Equal(MeasureOfAngle(CAB),35)",
"IsMidpointOfLine(D,AC)",
"IsMidpointOfLine(E,AB)"
] | [] | Value(MeasureOfAngle(ACF)) | 65 | [
"midsegment_of_triangle_judgment_midpoint(1,ED,ABC)",
"midsegment_of_triangle_property_parallel(1,ED,ABC)",
"parallel_property_corresponding_angle(1,ED,BC,A)",
"triangle_property_angle_sum(1,ABC)",
"adjacent_complementary_angle(1,BCD,DCF)"
] | {"START": ["midsegment_of_triangle_judgment_midpoint(1,ED,ABC)", "triangle_property_angle_sum(1,ABC)", "adjacent_complementary_angle(1,BCD,DCF)"], "midsegment_of_triangle_judgment_midpoint(1,ED,ABC)": ["midsegment_of_triangle_property_parallel(1,ED,ABC)"], "midsegment_of_triangle_property_parallel(1,ED,ABC)": ["paralle... | |
6,817 | ChenyangMao_2023-06-04 | GeoQA-angle-10032 | 5 | 如图所示，∠DAB=60°，BE是∠ABC的角平分线，四边形ABCD是▱。求∠BED的大小。 | As shown in the diagram, ∠DAB=60°, BE bisects ∠ABC, AD and BC are opposite sides of the ▱ ABCD. Find the measure of ∠BED. | 6817.png | [
"Shape(AB,BE,ED,DA)",
"Shape(CE,EB,BC)",
"Collinear(DEC)"
] | [
"Equal(MeasureOfAngle(DAB),60)",
"IsBisectorOfAngle(BE,ABC)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(BED)) | 120 | [
"parallel_property_ipsilateral_internal_angle(1,AD,BC)",
"parallel_property_ipsilateral_internal_angle(1,BA,CD)",
"angle_addition(1,ABE,EBC)",
"adjacent_complementary_angle(1,CEB,BED)",
"triangle_property_angle_sum(1,CEB)"
] | {"START": ["parallel_property_ipsilateral_internal_angle(1,AD,BC)", "parallel_property_ipsilateral_internal_angle(1,BA,CD)", "angle_addition(1,ABE,EBC)", "adjacent_complementary_angle(1,CEB,BED)", "triangle_property_angle_sum(1,CEB)"]} | |
6,818 | ChenyangMao_2023-06-04 | GeoQA-angle-10033 | 6 | 如图所示，BD=BC，∠BCD=55°，AD和BC是平行四边形ABCD的一组对边，BE⊥AE。求∠EAB的大小。 | As shown in the diagram, BD=BC, ∠BCD=55°, quadrilateral ABCD is a parallelogram, BE is perpendicular to AE. Find the measure of ∠EAB. | 6818.png | [
"Shape(DA,AE,ED)",
"Shape(EA,AB,BE)",
"Shape(CD,DE,EB,BC)",
"Collinear(DEB)"
] | [
"Equal(LengthOfLine(BD),LengthOfLine(BC))",
"Equal(MeasureOfAngle(BCD),55)",
"Parallelogram(ABCD)",
"PerpendicularBetweenLine(BE,AE)"
] | [] | Value(MeasureOfAngle(EAB)) | 35 | [
"isosceles_triangle_judgment_line_equal(1,BCD)",
"isosceles_triangle_property_angle_equal(1,BCD)",
"triangle_property_angle_sum(1,EAB)",
"triangle_property_angle_sum(1,CDB)",
"angle_addition(1,ABE,EBC)",
"parallel_property_ipsilateral_internal_angle(1,BA,CD)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,BCD)", "triangle_property_angle_sum(1,EAB)", "triangle_property_angle_sum(1,CDB)", "angle_addition(1,ABE,EBC)", "parallel_property_ipsilateral_internal_angle(1,BA,CD)"], "isosceles_triangle_judgment_line_equal(1,BCD)": ["isosceles_triangle_property_angle_equal(1,BCD)... | |
6,819 | ChenyangMao_2023-06-04 | GeoQA-angle-10034 | 2 | 如图所示，Add(MeasureOfAngle(BAD)=MeasureOfAngle(DCB))，BC和AD是平行四边形BADC的一组对边。求∠CBA的大小。 | As shown in the diagram, Add(MeasureOfAngle(BAD)=MeasureOfAngle(DCB)), AB and DC are opposite sides of the parallelogram BADC. Find the measure of ∠CBA. | 6819.png | [
"Shape(BA,AD,DC,CB)"
] | [
"Equal(Add(MeasureOfAngle(BAD),MeasureOfAngle(DCB)),130)",
"Parallelogram(BADC)"
] | [] | Value(MeasureOfAngle(CBA)) | 115 | [
"parallelogram_property_opposite_angle_equal(1,BADC)",
"quadrilateral_property_angle_sum(1,BADC)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,BADC)", "quadrilateral_property_angle_sum(1,BADC)"]} | |
6,820 | ChenyangMao_2023-06-04 | GeoQA-angle-10035 | 1 | 如图所示，∠ABC=70°，AD和BC是平行四边形ABCD的一组对边。求∠CDA的大小。 | As shown in the diagram, ∠ABC=70°, AD and BC are opposite sides of the parallelogram ABCD. Find the measure of ∠CDA. | 6820.png | [
"Shape(AB,BC,CD,DA)"
] | [
"Equal(MeasureOfAngle(ABC),70)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(CDA)) | 70 | [
"parallelogram_property_opposite_angle_equal(1,BCDA)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,BCDA)"]} | |
6,821 | ChenyangMao_2023-06-04 | GeoQA-angle-10036 | 2 | 如图所示，Add(MeasureOfAngle(DAB)=MeasureOfAngle(BCD))，四边形DABC是平行四边形。求∠ABC的大小。 | As shown in the diagram, Add(MeasureOfAngle(DAB)=MeasureOfAngle(BCD)), quadrilateral DABC is a ▱. Find the measure of ∠ABC. | 6821.png | [
"Shape(DA,AB,BC,CD)"
] | [
"Equal(Add(MeasureOfAngle(DAB),MeasureOfAngle(BCD)),160)",
"Parallelogram(DABC)"
] | [] | Value(MeasureOfAngle(ABC)) | 100 | [
"parallelogram_property_opposite_angle_equal(1,DABC)",
"quadrilateral_property_angle_sum(1,DABC)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,DABC)", "quadrilateral_property_angle_sum(1,DABC)"]} | |
6,822 | ChenyangMao_2023-06-04 | GeoQA-angle-10037 | 2 | 如图所示，Add(MeasureOfAngle(ABC)=MeasureOfAngle(CDA))，AD和BC是▱ABCD的一组对边。求∠DAB的大小。 | As shown in the diagram, Add(MeasureOfAngle(ABC)=MeasureOfAngle(CDA)), BA and CD are opposite sides of the ▱ ABCD. Find the measure of ∠DAB. | 6822.png | [
"Shape(AB,BC,CD,DA)"
] | [
"Equal(Add(MeasureOfAngle(ABC),MeasureOfAngle(CDA)),100)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(DAB)) | 130 | [
"parallelogram_property_opposite_angle_equal(1,ABCD)",
"quadrilateral_property_angle_sum(1,ABCD)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,ABCD)", "quadrilateral_property_angle_sum(1,ABCD)"]} | |
6,823 | ChenyangMao_2023-06-04 | GeoQA-angle-10038 | 2 | 如图所示，Add(MeasureOfAngle(ABC)=MeasureOfAngle(CDA))，四边形ABCD是▱。求∠BCD的大小。 | As shown in the diagram, Add(MeasureOfAngle(ABC)=MeasureOfAngle(CDA)), BA and CD are opposite sides of the parallelogram ABCD. Find the measure of ∠BCD. | 6823.png | [
"Shape(AB,BC,CD,DA)"
] | [
"Equal(Add(MeasureOfAngle(ABC),MeasureOfAngle(CDA)),140)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(BCD)) | 110 | [
"parallelogram_property_opposite_angle_equal(1,ABCD)",
"quadrilateral_property_angle_sum(1,ABCD)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,ABCD)", "quadrilateral_property_angle_sum(1,ABCD)"]} | |
6,824 | ChenyangMao_2023-06-04 | GeoQA-angle-10039 | 2 | 如图所示，∠DAB=115°，∠DBC=25°，ABCD是▱。求∠CDB的大小。 | As shown in the diagram, ∠DAB=115°, ∠DBC=25°, ABCD is a ▱. Find the measure of ∠CDB. | 6824.png | [
"Shape(AB,BD,DA)",
"Shape(DB,BC,CD)"
] | [
"Equal(MeasureOfAngle(DAB),115)",
"Equal(MeasureOfAngle(DBC),25)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(CDB)) | 40 | [
"triangle_property_angle_sum(1,DBC)",
"parallelogram_property_opposite_angle_equal(1,ABCD)"
] | {"START": ["triangle_property_angle_sum(1,DBC)", "parallelogram_property_opposite_angle_equal(1,ABCD)"]} | |
6,825 | ChenyangMao_2023-06-04 | GeoQA-angle-10040 | 1 | 如图所示，∠DAB=150°，ABCD是▱。求∠BCD的大小。 | As shown in the diagram, ∠DAB=150°, quadrilateral ABCD is a ▱. Find the measure of ∠BCD. | 6825.png | [
"Shape(AB,BC,CD,DA)"
] | [
"Equal(MeasureOfAngle(DAB),150)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(BCD)) | 150 | [
"parallelogram_property_opposite_angle_equal(1,ABCD)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,ABCD)"]} | |
6,826 | ChenyangMao_2023-06-04 | GeoQA-angle-10041 | 2 | 如图所示，Add(MeasureOfAngle(ABC)=MeasureOfAngle(CDA))，BA和CD是平行四边形ABCD的一组对边。求∠DAB的大小。 | As shown in the diagram, Add(MeasureOfAngle(ABC)=MeasureOfAngle(CDA)), BA and CD are opposite sides of the parallelogram ABCD. Find the measure of ∠DAB. | 6826.png | [
"Shape(AB,BC,CD,DA)"
] | [
"Equal(Add(MeasureOfAngle(ABC),MeasureOfAngle(CDA)),200)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(DAB)) | 80 | [
"parallelogram_property_opposite_angle_equal(1,ABCD)",
"quadrilateral_property_angle_sum(1,ABCD)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,ABCD)", "quadrilateral_property_angle_sum(1,ABCD)"]} | |
6,827 | ChenyangMao_2023-06-04 | GeoQA-angle-10042 | 9 | 如图所示，AB=BD，CE=CB，∠BDA=65°，BA和CD是平行四边形ABCD的一组对边。求∠ECD的大小。 | As shown in the diagram, AB=BD, CE=CB, ∠BDA=65°, ABCD is a ▱. Find the measure of ∠ECD. | 6827.png | [
"Shape(AB,BE,ED,DA)",
"Shape(DE,EC,CD)",
"Shape(CE,EB,BC)",
"Collinear(DEB)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(BD))",
"Equal(LengthOfLine(CE),LengthOfLine(CB))",
"Equal(MeasureOfAngle(BDA),65)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(ECD)) | 15 | [
"parallel_property_alternate_interior_angle(2,AD,BC)",
"isosceles_triangle_judgment_line_equal(1,BDA)",
"isosceles_triangle_judgment_line_equal(1,CEB)",
"isosceles_triangle_property_angle_equal(1,BDA)",
"isosceles_triangle_property_angle_equal(1,CEB)",
"angle_addition(1,BCE,ECD)",
"parallel_property_ips... | {"START": ["parallel_property_alternate_interior_angle(2,AD,BC)", "isosceles_triangle_judgment_line_equal(1,BDA)", "isosceles_triangle_judgment_line_equal(1,CEB)", "angle_addition(1,BCE,ECD)", "parallel_property_ipsilateral_internal_angle(1,AD,BC)", "parallel_property_ipsilateral_internal_angle(1,BA,CD)", "triangle_pro... | |
6,828 | ChenyangMao_2023-06-04 | GeoQA-angle-10043 | 2 | 如图所示，∠CDA=120°，BA和CD是平行四边形ABCD的一组对边。求∠CBM的大小。 | As shown in the diagram, ∠CDA=120°, BA and CD are opposite sides of the ▱ ABCD. Find the measure of ∠CBM. | 6828.png | [
"Shape(AB,BC,CD,DA)",
"Shape(CB,BM)",
"Collinear(ABM)"
] | [
"Equal(MeasureOfAngle(CDA),120)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(CBM)) | 60 | [
"adjacent_complementary_angle(1,ABC,CBM)",
"parallelogram_property_opposite_angle_equal(1,BCDA)"
] | {"START": ["adjacent_complementary_angle(1,ABC,CBM)", "parallelogram_property_opposite_angle_equal(1,BCDA)"]} | |
6,829 | ChenyangMao_2023-06-04 | GeoQA-angle-10044 | 2 | 如图所示，Add(MeasureOfAngle(DAB)=MeasureOfAngle(BCD))，BA和CD是▱ABCD的一组对边。求∠ABC的大小。 | As shown in the diagram, Add(MeasureOfAngle(DAB)=MeasureOfAngle(BCD)), ABCD is a parallelogram. Find the measure of ∠ABC. | 6829.png | [
"Shape(AB,BC,CD,DA)"
] | [
"Equal(Add(MeasureOfAngle(DAB),MeasureOfAngle(BCD)),110)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(ABC)) | 125 | [
"parallelogram_property_opposite_angle_equal(1,BCDA)",
"quadrilateral_property_angle_sum(1,ABCD)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,BCDA)", "quadrilateral_property_angle_sum(1,ABCD)"]} | |
6,830 | ChenyangMao_2023-06-04 | GeoQA-angle-10045 | 4 | 如图所示，AE=DE，∠DEA=46°，ADCB是▱。求∠CBA的大小。 | As shown in the diagram, AE=DE, ∠DEA=46°, AB and DC are opposite sides of the parallelogram ADCB. Find the measure of ∠CBA. | 6830.png | [
"Shape(AD,DC,CB,BA)",
"Shape(AD,DC,CE,EA)",
"Collinear(DCE)"
] | [
"Equal(LengthOfLine(AE),LengthOfLine(DE))",
"Equal(MeasureOfAngle(DEA),46)",
"Parallelogram(ADCB)"
] | [] | Value(MeasureOfAngle(CBA)) | 67 | [
"triangle_property_angle_sum(1,ADE)",
"isosceles_triangle_judgment_line_equal(1,EAD)",
"isosceles_triangle_property_angle_equal(1,EAD)",
"parallelogram_property_opposite_angle_equal(1,DCBA)"
] | {"START": ["triangle_property_angle_sum(1,ADE)", "isosceles_triangle_judgment_line_equal(1,EAD)", "parallelogram_property_opposite_angle_equal(1,DCBA)"], "isosceles_triangle_judgment_line_equal(1,EAD)": ["isosceles_triangle_property_angle_equal(1,EAD)"]} | |
6,831 | ChenyangMao_2023-06-04 | GeoQA-angle-10046 | 3 | 如图所示，Mul(MeasureOfAngle(DAB)=1)，四边形ABCD是平行四边形。求∠CDA的大小。 | As shown in the diagram, Mul(MeasureOfAngle(DAB)=1), BA and CD are opposite sides of the ▱ ABCD. Find the measure of ∠CDA. | 6831.png | [
"Shape(AB,BC,CD,DA)"
] | [
"Equal(Mul(MeasureOfAngle(DAB),1),Mul(MeasureOfAngle(ABC),2))",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(CDA)) | 60 | [
"parallelogram_property_opposite_angle_equal(1,ABCD)",
"parallelogram_property_opposite_angle_equal(1,BCDA)",
"quadrilateral_property_angle_sum(1,ABCD)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,ABCD)", "parallelogram_property_opposite_angle_equal(1,BCDA)", "quadrilateral_property_angle_sum(1,ABCD)"]} | |
6,832 | ChenyangMao_2023-06-04 | GeoQA-angle-10047 | 2 | 如图所示，Add(MeasureOfAngle(BAD)=MeasureOfAngle(DCB))，BADC是▱。求∠ADC的大小。 | As shown in the diagram, Add(MeasureOfAngle(BAD)=MeasureOfAngle(DCB)), AB and DC are opposite sides of the parallelogram BADC. Find the measure of ∠ADC. | 6832.png | [
"Shape(BA,AD,DC,CB)"
] | [
"Equal(Add(MeasureOfAngle(BAD),MeasureOfAngle(DCB)),120)",
"Parallelogram(BADC)"
] | [] | Value(MeasureOfAngle(ADC)) | 120 | [
"parallelogram_property_opposite_angle_equal(1,BADC)",
"quadrilateral_property_angle_sum(1,BADC)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,BADC)", "quadrilateral_property_angle_sum(1,BADC)"]} | |
6,833 | ChenyangMao_2023-06-04 | GeoQA-angle-10049 | 2 | 如图所示，∠DAB=110°，BA和CD是▱ABCD的一组对边。求∠DCE的大小。 | As shown in the diagram, ∠DAB=110°, BA and CD are opposite sides of the parallelogram ABCD. Find the measure of ∠DCE. | 6833.png | [
"Shape(AB,BC,CD,DA)",
"Shape(DC,CE)",
"Collinear(BCE)"
] | [
"Equal(MeasureOfAngle(DAB),110)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(DCE)) | 70 | [
"parallelogram_property_opposite_angle_equal(1,ABCD)",
"adjacent_complementary_angle(1,BCD,DCE)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,ABCD)", "adjacent_complementary_angle(1,BCD,DCE)"]} | |
6,834 | ChenyangMao_2023-06-04 | GeoQA-angle-10050 | 5 | 如图所示，∠CDA=58°，AE是∠DAB的角平分线，AD和BC是▱ABCD的一组对边。求∠AEC的大小。 | As shown in the diagram, ∠CDA=58°, AE bisects ∠DAB, quadrilateral ABCD is a parallelogram. Find the measure of ∠AEC. | 6834.png | [
"Shape(AB,BE,EA)",
"Shape(AE,EC,CD,DA)",
"Collinear(BEC)"
] | [
"Equal(MeasureOfAngle(CDA),58)",
"IsBisectorOfAngle(AE,DAB)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(AEC)) | 119 | [
"parallelogram_property_opposite_angle_equal(1,BCDA)",
"parallel_property_ipsilateral_internal_angle(1,DC,AB)",
"triangle_property_angle_sum(1,ABE)",
"angle_addition(1,DAE,EAB)",
"adjacent_complementary_angle(1,BEA,AEC)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,BCDA)", "parallel_property_ipsilateral_internal_angle(1,DC,AB)", "triangle_property_angle_sum(1,ABE)", "angle_addition(1,DAE,EAB)", "adjacent_complementary_angle(1,BEA,AEC)"]} | |
6,835 | ChenyangMao_2023-06-04 | GeoQA-angle-10051 | 4 | 如图所示，AB=AC，∠BAC=40°，EDCB是平行四边形。求∠BED的大小。 | As shown in the diagram, AB=AC, ∠BAC=40°, quadrilateral EDCB is a ▱. Find the measure of ∠BED. | 6835.png | [
"Shape(BA,AD,DC,CB)",
"Shape(BE,ED,DC,CB)",
"Collinear(ADC)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(AC))",
"Equal(MeasureOfAngle(BAC),40)",
"Parallelogram(EDCB)"
] | [] | Value(MeasureOfAngle(BED)) | 70 | [
"isosceles_triangle_judgment_line_equal(1,ACB)",
"isosceles_triangle_property_angle_equal(1,ACB)",
"triangle_property_angle_sum(1,BAC)",
"parallelogram_property_opposite_angle_equal(1,EDCB)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,ACB)", "triangle_property_angle_sum(1,BAC)", "parallelogram_property_opposite_angle_equal(1,EDCB)"], "isosceles_triangle_judgment_line_equal(1,ACB)": ["isosceles_triangle_property_angle_equal(1,ACB)"]} | |
6,836 | ChenyangMao_2023-06-04 | GeoQA-angle-10052 | 2 | 如图所示，∠DCE=80°，BA和CD是平行四边形ABCD的一组对边。求∠DAB的大小。 | As shown in the diagram, ∠DCE=80°, AD and BC are opposite sides of the ▱ ABCD. Find the measure of ∠DAB. | 6836.png | [
"Shape(AB,BC,CD,DA)",
"Shape(DC,CE)",
"Collinear(BCE)"
] | [
"Equal(MeasureOfAngle(DCE),80)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(DAB)) | 100 | [
"parallelogram_property_opposite_angle_equal(1,ABCD)",
"adjacent_complementary_angle(1,BCD,DCE)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,ABCD)", "adjacent_complementary_angle(1,BCD,DCE)"]} | |
6,837 | ChenyangMao_2023-06-04 | GeoQA-angle-10053 | 3 | 如图所示，∠DAB=60°，CDAB是▱，BF⊥DF，DE⊥CE。求∠FHE的大小。 | As shown in the diagram, ∠DAB=60°, DC and AB are opposite sides of the ▱ CDAB, BF⊥DF, DE is perpendicular to CE. Find the measure of ∠FHE. | 6837.png | [
"Shape(CF,FH,HE,EC)",
"Shape(BE,EH,HB)",
"Shape(HF,FD,DH)",
"Shape(BH,HD,DA,AB)",
"Collinear(CEB)",
"Collinear(CFD)",
"Collinear(EHD)",
"Collinear(BHF)"
] | [
"Equal(MeasureOfAngle(DAB),60)",
"Parallelogram(CDAB)",
"PerpendicularBetweenLine(BF,DF)",
"PerpendicularBetweenLine(DE,CE)"
] | [] | Value(MeasureOfAngle(FHE)) | 120 | [
"parallelogram_property_opposite_angle_equal(1,CDAB)",
"adjacent_complementary_angle(1,CFH,HFD)",
"quadrilateral_property_angle_sum(1,CFHE)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,CDAB)", "adjacent_complementary_angle(1,CFH,HFD)", "quadrilateral_property_angle_sum(1,CFHE)"]} | |
6,838 | ChenyangMao_2023-06-04 | GeoQA-angle-10054 | 2 | 如图所示，∠DAB=100°，AD和BC是▱ABCD的一组对边。求∠DCE的大小。 | As shown in the diagram, ∠DAB=100°, AD and BC are opposite sides of the ▱ ABCD. Find the measure of ∠DCE. | 6838.png | [
"Shape(AB,BC,CD,DA)",
"Shape(DC,CE)",
"Collinear(BCE)"
] | [
"Equal(MeasureOfAngle(DAB),100)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(DCE)) | 80 | [
"adjacent_complementary_angle(1,BCD,DCE)",
"parallelogram_property_opposite_angle_equal(1,ABCD)"
] | {"START": ["adjacent_complementary_angle(1,BCD,DCE)", "parallelogram_property_opposite_angle_equal(1,ABCD)"]} | |
6,839 | ChenyangMao_2023-06-04 | GeoQA-angle-10055 | 11 | 如图所示，∠OAB=80°，OD平分∠FOC，F平分线段AD，BA和CD是▱ABCD的一组对边。求∠ABD的大小。 | As shown in the diagram, ∠OAB=80°, OD is the angle bisector of ∠FOC, F is the midpoint of segment AD, AD and BC are opposite sides of the parallelogram ABCD. Find the measure of ∠ABD. | 6839.png | [
"Shape(AB,BO,OA)",
"Shape(FA,AO,OF)",
"Shape(FO,OD,DF)",
"Shape(DO,OC,CD)",
"Shape(OB,BC,CO)",
"Collinear(AOC)",
"Collinear(BOD)",
"Collinear(AFD)"
] | [
"Equal(MeasureOfAngle(OAB),80)",
"IsBisectorOfAngle(OD,FOC)",
"IsMidpointOfLine(F,AD)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(ABD)) | 50 | [
"parallelogram_property_diagonal_bisection(1,ABCD,O)",
"parallelogram_property_diagonal_bisection(1,BCDA,O)",
"parallel_property_alternate_interior_angle(2,BA,CD)",
"midsegment_of_triangle_judgment_midpoint(1,FO,DAB)",
"midsegment_of_triangle_judgment_midpoint(1,OF,ACD)",
"midsegment_of_triangle_property_... | {"START": ["parallelogram_property_diagonal_bisection(1,ABCD,O)", "parallelogram_property_diagonal_bisection(1,BCDA,O)", "parallel_property_alternate_interior_angle(2,BA,CD)", "adjacent_complementary_angle(1,AOF,FOC)", "angle_addition(1,FOD,DOC)"], "midsegment_of_triangle_judgment_midpoint(1,FO,DAB)": ["midsegment_of_t... | |
6,840 | DengfengYue_2023-06-04 | GeoQA-angle-10056 | 9 | 如图所示，∠DBE=70°，∠ECF=50°，D是线段AB的中点，E是线段BC的中点，F是线段AC的中点。求∠DEF的大小。 | As shown in the diagram, ∠DBE=70°, ∠ECF=50°, D is the midpoint of segment AB, E is the midpoint of segment BC, F is the midpoint of segment AC. Find the measure of ∠DEF. | 6840.png | [
"Shape(AD,DE,EF,FA)",
"Shape(DB,BE,ED)",
"Shape(FE,EC,CF)",
"Collinear(ADB)",
"Collinear(AFC)",
"Collinear(BEC)"
] | [
"Equal(MeasureOfAngle(DBE),70)",
"Equal(MeasureOfAngle(ECF),50)",
"IsMidpointOfLine(D,AB)",
"IsMidpointOfLine(E,BC)",
"IsMidpointOfLine(F,AC)"
] | [] | Value(MeasureOfAngle(DEF)) | 60 | [
"triangle_property_angle_sum(1,BCA)",
"midsegment_of_triangle_judgment_midpoint(1,ED,BCA)",
"midsegment_of_triangle_judgment_midpoint(1,FE,CAB)",
"midsegment_of_triangle_property_parallel(1,ED,BCA)",
"midsegment_of_triangle_property_parallel(1,FE,CAB)",
"parallel_property_collinear_extend(3,AC,DE,F)",
"... | {"START": ["triangle_property_angle_sum(1,BCA)", "midsegment_of_triangle_judgment_midpoint(1,ED,BCA)", "midsegment_of_triangle_judgment_midpoint(1,FE,CAB)"], "midsegment_of_triangle_judgment_midpoint(1,ED,BCA)": ["midsegment_of_triangle_property_parallel(1,ED,BCA)"], "midsegment_of_triangle_judgment_midpoint(1,FE,CAB)"... | |
6,841 | DengfengYue_2023-06-04 | GeoQA-angle-10057 | 2 | 如图所示，Add(MeasureOfAngle(DAB)=MeasureOfAngle(BCD))，BA和CD是平行四边形ABCD的一组对边。求∠ABC的大小。 | As shown in the diagram, Add(MeasureOfAngle(DAB)=MeasureOfAngle(BCD)), BA and CD are opposite sides of the ▱ ABCD. Find the measure of ∠ABC. | 6841.png | [
"Shape(DA,AB,BC,CD)"
] | [
"Equal(Add(MeasureOfAngle(DAB),MeasureOfAngle(BCD)),140)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(ABC)) | 110 | [
"parallelogram_property_opposite_angle_equal(1,ABCD)",
"parallel_property_ipsilateral_internal_angle(1,BA,CD)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,ABCD)", "parallel_property_ipsilateral_internal_angle(1,BA,CD)"]} | |
6,842 | DengfengYue_2023-06-04 | GeoQA-angle-10058 | 12 | 如图所示，AB=CD，∠BDC=44°，∠DBA=100°，E是线段BC的中点，F是线段AD的中点，G平分线段BD。求∠GEF的大小。 | As shown in the diagram, AB=CD, ∠BDC=44°, ∠DBA=100°, E is the midpoint of segment BC, F bisects segment AD, G is the midpoint of segment BD. Find the measure of ∠GEF. | 6842.png | [
"Shape(BA,AF,FH,HB)",
"Shape(BH,HE,EB)",
"Shape(EH,HG,GE)",
"Shape(CE,EG,GD,DC)",
"Shape(GH,HF,FG)",
"Shape(GF,FD,DG)",
"Collinear(BEC)",
"Collinear(BHGD)",
"Collinear(EHF)",
"Collinear(AFD)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(CD))",
"Equal(MeasureOfAngle(BDC),44)",
"Equal(MeasureOfAngle(DBA),100)",
"IsMidpointOfLine(E,BC)",
"IsMidpointOfLine(F,AD)",
"IsMidpointOfLine(G,BD)"
] | [] | Value(MeasureOfAngle(GEF)) | 28 | [
"midsegment_of_triangle_judgment_midpoint(1,GF,DBA)",
"midsegment_of_triangle_judgment_midpoint(1,GE,BDC)",
"midsegment_of_triangle_property_parallel(1,GF,DBA)",
"midsegment_of_triangle_property_parallel(1,GE,BDC)",
"midsegment_of_triangle_property_length(1,GF,DBA)",
"midsegment_of_triangle_property_lengt... | {"START": ["midsegment_of_triangle_judgment_midpoint(1,GF,DBA)", "midsegment_of_triangle_judgment_midpoint(1,GE,BDC)", "angle_addition(1,FGH,HGE)", "triangle_property_angle_sum(1,EFG)"], "isosceles_triangle_judgment_line_equal(1,GEF)": ["isosceles_triangle_property_angle_equal(1,GEF)"], "midsegment_of_triangle_judgment... | |
6,843 | DengfengYue_2023-06-04 | GeoQA-angle-10059 | 4 | 如图所示，AB=AC，∠CAB=40°，BA和CD是▱ABCD的一组对边。求∠CDA的大小。 | As shown in the diagram, AB=AC, ∠CAB=40°, BA and CD are opposite sides of the parallelogram ABCD. Find the measure of ∠CDA. | 6843.png | [
"Shape(DA,AC,CD)",
"Shape(CA,AB,BC)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(AC))",
"Equal(MeasureOfAngle(CAB),40)",
"Parallelogram(ABCD)"
] | [] | Value(MeasureOfAngle(CDA)) | 70 | [
"isosceles_triangle_judgment_line_equal(1,ABC)",
"isosceles_triangle_property_angle_equal(1,ABC)",
"triangle_property_angle_sum(1,CAB)",
"parallelogram_property_opposite_angle_equal(1,BCDA)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,ABC)", "triangle_property_angle_sum(1,CAB)", "parallelogram_property_opposite_angle_equal(1,BCDA)"], "isosceles_triangle_judgment_line_equal(1,ABC)": ["isosceles_triangle_property_angle_equal(1,ABC)"]} | |
6,844 | DengfengYue_2023-06-04 | GeoQA-angle-10060 | 2 | 如图所示，∠DGJ=90°，∠EFD=90°，∠FDE=45°。求∠FEG的大小减去∠EFD的大小。 | As shown in the diagram, ∠DGJ=90°, ∠EFD=90°, ∠FDE=45°. Find the measure of ∠FEG minus the measure of ∠EFD. | 6844.png | [
"Shape(AB,BH,HG,GE,EA)",
"Shape(DE,EF,FD)",
"Shape(FE,EG,GH,HI,IF)",
"Shape(FI,IC,CF)",
"Shape(HJ,JI,IH)",
"Collinear(AEFC)",
"Collinear(DFIJ)",
"Collinear(DEG)",
"Collinear(BHIC)"
] | [
"Equal(MeasureOfAngle(DGJ),90)",
"Equal(MeasureOfAngle(EFD),90)",
"Equal(MeasureOfAngle(FDE),45)"
] | [] | Value(Sub(MeasureOfAngle(FEG),MeasureOfAngle(EFD))) | 45 | [
"triangle_property_angle_sum(1,DEF)",
"adjacent_complementary_angle(1,DEF,FEG)"
] | {"START": ["triangle_property_angle_sum(1,DEF)", "adjacent_complementary_angle(1,DEF,FEG)"]} | |
6,845 | DengfengYue_2023-06-04 | GeoQA-angle-10061 | 3 | 如图所示，∠BAC=30°。求∠DBC的大小与∠BCE的大小之和。 | As shown in the diagram, ∠BAC=30°. Find the sum of the measure of ∠DBC and the measure of ∠BCE. | 6845.png | [
"Shape(BA,AC,CB)",
"Shape(BC,CE,ED,DB)",
"Collinear(ABD)",
"Collinear(ACE)"
] | [
"Equal(MeasureOfAngle(BAC),30)"
] | [] | Value(Add(MeasureOfAngle(DBC),MeasureOfAngle(BCE))) | 210 | [
"triangle_property_angle_sum(1,BAC)",
"adjacent_complementary_angle(1,DBC,CBA)",
"adjacent_complementary_angle(1,ACB,BCE)"
] | {"START": ["triangle_property_angle_sum(1,BAC)", "adjacent_complementary_angle(1,DBC,CBA)", "adjacent_complementary_angle(1,ACB,BCE)"]} | |
6,846 | DengfengYue_2023-06-04 | GeoQA-angle-10063 | 4 | 如图所示，Mul(1/4=MeasureOfAngle(ABC))，Mul(2=MeasureOfAngle(ABC))，AD平分∠CAB，AE⊥CE。求∠ECA的大小。 | As shown in the diagram, Mul(1/4=MeasureOfAngle(ABC)), Mul(2=MeasureOfAngle(ABC)), AD bisects ∠CAB, AE⊥CE. Find the measure of ∠ECA. | 6846.png | [
"Shape(AB,BD,DA)",
"Shape(AD,DE,EA)",
"Shape(AE,EC,CA)",
"Collinear(BDEC)"
] | [
"Equal(Mul(1/4,MeasureOfAngle(ABC)),MeasureOfAngle(EAD))",
"Equal(Mul(2,MeasureOfAngle(ABC)),MeasureOfAngle(CAB))",
"IsBisectorOfAngle(AD,CAB)",
"PerpendicularBetweenLine(AE,CE)"
] | [] | Value(MeasureOfAngle(ECA)) | 60 | [
"triangle_property_angle_sum(1,ABC)",
"triangle_property_angle_sum(1,AEC)",
"angle_addition(1,CAD,DAB)",
"angle_addition(1,CAE,EAD)"
] | {"START": ["triangle_property_angle_sum(1,ABC)", "triangle_property_angle_sum(1,AEC)", "angle_addition(1,CAD,DAB)", "angle_addition(1,CAE,EAD)"]} | |
6,847 | DengfengYue_2023-06-04 | GeoQA-angle-10064 | 4 | 如图所示，∠DAE=20°，∠ECD=35°，∠FBE=45°。求∠DFE的大小。 | As shown in the diagram, ∠DAE=20°, ∠ECD=35°, ∠FBE=45°. Find the measure of ∠DFE. | 6847.png | [
"Shape(AF,FD,DA)",
"Shape(FB,BE,EF)",
"Shape(DF,FE,EC,CD)",
"Collinear(AFE)",
"Collinear(ADC)",
"Collinear(DFB)",
"Collinear(BEC)"
] | [
"Equal(MeasureOfAngle(DAE),20)",
"Equal(MeasureOfAngle(ECD),35)",
"Equal(MeasureOfAngle(FBE),45)"
] | [] | Value(MeasureOfAngle(DFE)) | 100 | [
"triangle_property_angle_sum(1,FBE)",
"triangle_property_angle_sum(1,AEC)",
"adjacent_complementary_angle(1,DFE,EFB)",
"adjacent_complementary_angle(1,BEF,FEC)"
] | {"START": ["triangle_property_angle_sum(1,FBE)", "triangle_property_angle_sum(1,AEC)", "adjacent_complementary_angle(1,DFE,EFB)", "adjacent_complementary_angle(1,BEF,FEC)"]} | |
6,848 | DengfengYue_2023-06-04 | GeoQA-angle-10066 | 2 | 如图所示，∠ABD=30°，∠BDA=80°，BD是∠ABC的角平分线。求∠BCD的大小。 | As shown in the diagram, ∠ABD=30°, ∠BDA=80°, BD bisects ∠ABC. Find the measure of ∠BCD. | 6848.png | [
"Shape(AB,BD,DA)",
"Shape(DB,BC,CD)",
"Collinear(ADC)"
] | [
"Equal(MeasureOfAngle(ABD),30)",
"Equal(MeasureOfAngle(BDA),80)",
"IsBisectorOfAngle(BD,ABC)"
] | [] | Value(MeasureOfAngle(BCD)) | 50 | [
"adjacent_complementary_angle(1,CDB,BDA)",
"triangle_property_angle_sum(1,DBC)"
] | {"START": ["adjacent_complementary_angle(1,CDB,BDA)", "triangle_property_angle_sum(1,DBC)"]} | |
6,849 | DengfengYue_2023-06-04 | GeoQA-angle-10068 | 2 | 如图所示，∠ABP=20°，∠ACP=50°，BP是∠ABC的角平分线，CP是∠ACM的角平分线。求∠CPD的大小。 | As shown in the diagram, ∠ABP=20°, ∠ACP=50°, BP is the angle bisector of ∠ABC, CP bisects ∠ACM. Find the measure of ∠CPD. | 6849.png | [
"Shape(AB,BD,DA)",
"Shape(PD,DC,CP)",
"Shape(BC,CD,DB)",
"Collinear(ADC)",
"Collinear(BDP)",
"Collinear(BCM)"
] | [
"Equal(MeasureOfAngle(ABP),20)",
"Equal(MeasureOfAngle(ACP),50)",
"IsBisectorOfAngle(BP,ABC)",
"IsBisectorOfAngle(CP,ACM)"
] | [] | Value(MeasureOfAngle(CPD)) | 30 | [
"adjacent_complementary_angle(1,BCP,PCM)",
"triangle_property_angle_sum(1,CPB)"
] | {"START": ["adjacent_complementary_angle(1,BCP,PCM)", "triangle_property_angle_sum(1,CPB)"]} | |
6,850 | DengfengYue_2023-06-04 | GeoQA-angle-10070 | 9 | 如图所示，∠CAB=50°，∠CPB=120°，BD平分∠ABP，CE是∠PCA的角平分线。求∠CFB的大小。 | As shown in the diagram, ∠CAB=50°, ∠CPB=120°, BD bisects ∠ABP, CE is the angle bisector of ∠PCA. Find the measure of ∠CFB. | 6850.png | [
"Shape(AE,EF,FD,DA)",
"Shape(EB,BF,FE)",
"Shape(DF,FC,CD)",
"Shape(CF,FB,BP,PC)",
"Shape(BC,CP,PB)",
"Collinear(AEB)",
"Collinear(ADC)",
"Collinear(EFC)",
"Collinear(DFB)"
] | [
"Equal(MeasureOfAngle(CAB),50)",
"Equal(MeasureOfAngle(CPB),120)",
"IsBisectorOfAngle(BD,ABP)",
"IsBisectorOfAngle(CE,PCA)"
] | [] | Value(MeasureOfAngle(CFB)) | 85 | [
"triangle_property_angle_sum(1,BCP)",
"triangle_property_angle_sum(1,CFB)",
"triangle_property_angle_sum(1,ABC)",
"angle_addition(1,ABP,PBC)",
"angle_addition(1,BCP,PCA)",
"angle_addition(1,EBF,FBP)",
"angle_addition(1,PCF,FCD)",
"angle_addition(1,FBP,PBC)",
"angle_addition(1,BCP,PCF)"
] | {"START": ["triangle_property_angle_sum(1,BCP)", "triangle_property_angle_sum(1,CFB)", "triangle_property_angle_sum(1,ABC)", "angle_addition(1,ABP,PBC)", "angle_addition(1,BCP,PCA)", "angle_addition(1,EBF,FBP)", "angle_addition(1,PCF,FCD)", "angle_addition(1,FBP,PBC)", "angle_addition(1,BCP,PCF)"]} | |
6,851 | DengfengYue_2023-06-04 | GeoQA-angle-10071 | 1 | 如图所示，∠ABC=60°，∠CAB=55°。求∠BCA的大小。 | As shown in the diagram, ∠ABC=60°, ∠CAB=55°. Find the measure of ∠BCA. | 6851.png | [
"Shape(AB,BC,CA)"
] | [
"Equal(MeasureOfAngle(ABC),60)",
"Equal(MeasureOfAngle(CAB),55)"
] | [] | Value(MeasureOfAngle(BCA)) | 65 | [
"triangle_property_angle_sum(1,ABC)"
] | {"START": ["triangle_property_angle_sum(1,ABC)"]} | |
6,852 | DengfengYue_2023-06-04 | GeoQA-angle-10072 | 7 | 如图所示，∠CAB=60°。求∠BDE的大小、∠DEB的大小、∠CFG的大小与∠FGC的大小之和。 | As shown in the diagram, ∠CAB=60°. Find the sum of the measure of ∠BDE, the measure of ∠DEB, the measure of ∠CFG and the measure of ∠FGC. | 6852.png | [
"Shape(DE,EB,BD)",
"Shape(AB,BC,CA)",
"Shape(CF,FG,GC)",
"Collinear(DBCG)",
"Collinear(ABE)",
"Collinear(ACF)"
] | [
"Equal(MeasureOfAngle(CAB),60)"
] | [] | Value(Add(MeasureOfAngle(BDE),MeasureOfAngle(DEB),MeasureOfAngle(CFG),MeasureOfAngle(FGC))) | 240 | [
"triangle_property_angle_sum(1,DEB)",
"triangle_property_angle_sum(1,ABC)",
"triangle_property_angle_sum(1,CFG)",
"adjacent_complementary_angle(1,CBE,EBD)",
"adjacent_complementary_angle(1,DCA,ACG)",
"adjacent_complementary_angle(1,ABC,CBE)",
"adjacent_complementary_angle(1,ACG,GCF)"
] | {"START": ["triangle_property_angle_sum(1,DEB)", "triangle_property_angle_sum(1,ABC)", "triangle_property_angle_sum(1,CFG)", "adjacent_complementary_angle(1,CBE,EBD)", "adjacent_complementary_angle(1,DCA,ACG)", "adjacent_complementary_angle(1,ABC,CBE)", "adjacent_complementary_angle(1,ACG,GCF)"]} | |
6,853 | DengfengYue_2023-06-04 | GeoQA-angle-10075 | 2 | 如图所示，∠ABC=56°，∠ACD=120°。求∠CAB的大小。 | As shown in the diagram, ∠ABC=56°, ∠ACD=120°. Find the measure of ∠CAB. | 6853.png | [
"Shape(AB,BC,CA)",
"Collinear(BCD)"
] | [
"Equal(MeasureOfAngle(ABC),56)",
"Equal(MeasureOfAngle(ACD),120)"
] | [] | Value(MeasureOfAngle(CAB)) | 64 | [
"adjacent_complementary_angle(1,BCA,ACD)",
"triangle_property_angle_sum(1,ABC)"
] | {"START": ["adjacent_complementary_angle(1,BCA,ACD)", "triangle_property_angle_sum(1,ABC)"]} | |
6,854 | DengfengYue_2023-06-04 | GeoQA-angle-10076 | 4 | 如图所示，∠BAC=∠DBA，∠CBD=54°，CD平分∠ACB，BD垂直于CD。求∠BAC的大小。 | As shown in the diagram, ∠BAC=∠DBA, ∠CBD=54°, CD bisects ∠ACB, BD is perpendicular to CD. Find the measure of ∠BAC. | 6854.png | [
"Shape(BA,AC,CD,DB)",
"Shape(BD,DC,CB)"
] | [
"Equal(MeasureOfAngle(BAC),MeasureOfAngle(DBA))",
"Equal(MeasureOfAngle(CBD),54)",
"IsBisectorOfAngle(CD,ACB)",
"PerpendicularBetweenLine(BD,CD)"
] | [] | Value(MeasureOfAngle(BAC)) | 27 | [
"triangle_property_angle_sum(1,BDC)",
"angle_addition(1,ACD,DCB)",
"angle_addition(1,CBD,DBA)",
"triangle_property_angle_sum(1,BAC)"
] | {"START": ["triangle_property_angle_sum(1,BDC)", "angle_addition(1,ACD,DCB)", "angle_addition(1,CBD,DBA)", "triangle_property_angle_sum(1,BAC)"]} | |
6,855 | DengfengYue_2023-06-04 | GeoQA-angle-10077 | 4 | 如图所示，∠CAB=110°，BO平分∠ABC，CO平分∠BCA。求∠COB的大小。 | As shown in the diagram, ∠CAB=110°, BO bisects ∠ABC, CO is the angle bisector of ∠BCA. Find the measure of ∠COB. | 6855.png | [
"Shape(AB,BO,OC,CA)",
"Shape(OB,BC,CO)"
] | [
"Equal(MeasureOfAngle(CAB),110)",
"IsBisectorOfAngle(BO,ABC)",
"IsBisectorOfAngle(CO,BCA)"
] | [] | Value(MeasureOfAngle(COB)) | 145 | [
"triangle_property_angle_sum(1,ABC)",
"angle_addition(1,ABO,OBC)",
"angle_addition(1,BCO,OCA)",
"triangle_property_angle_sum(1,OBC)"
] | {"START": ["triangle_property_angle_sum(1,ABC)", "angle_addition(1,ABO,OBC)", "angle_addition(1,BCO,OCA)", "triangle_property_angle_sum(1,OBC)"]} | |
6,856 | DengfengYue_2023-06-04 | GeoQA-angle-10079 | 3 | 如图所示，∠BAC=90°，∠CBA=45°，∠CBD=30°。求∠DEA的大小。 | As shown in the diagram, ∠BAC=90°, ∠CBA=45°, ∠CBD=30°. Find the measure of ∠DEA. | 6856.png | [
"Shape(DC,CE,ED)",
"Shape(AE,EB,BA)",
"Shape(EC,CB,BE)",
"Collinear(DEB)",
"Collinear(AEC)"
] | [
"Equal(MeasureOfAngle(BAC),90)",
"Equal(MeasureOfAngle(CBA),45)",
"Equal(MeasureOfAngle(CBD),30)"
] | [] | Value(MeasureOfAngle(DEA)) | 105 | [
"angle_addition(1,CBE,EBA)",
"triangle_property_angle_sum(1,AEB)",
"adjacent_complementary_angle(1,DEA,AEB)"
] | {"START": ["angle_addition(1,CBE,EBA)", "triangle_property_angle_sum(1,AEB)", "adjacent_complementary_angle(1,DEA,AEB)"]} | |
6,857 | DengfengYue_2023-06-04 | GeoQA-angle-10080 | 3 | 如图所示，∠CDE=45°，∠ECD=90°，∠EDF=20°，∠FDG=90°。求∠GDB的大小。 | As shown in the diagram, ∠CDE=45°, ∠ECD=90°, ∠EDF=20°, ∠FDG=90°. Find the measure of ∠GDB. | 6857.png | [
"Shape(EC,CD,DE)",
"Shape(FD,DG,GF)",
"Shape(ED,DF,FE)",
"Collinear(CDB)"
] | [
"Equal(MeasureOfAngle(CDE),45)",
"Equal(MeasureOfAngle(ECD),90)",
"Equal(MeasureOfAngle(EDF),20)",
"Equal(MeasureOfAngle(FDG),90)"
] | [] | Value(MeasureOfAngle(GDB)) | 25 | [
"angle_addition(1,CDE,EDF)",
"angle_addition(1,CDF,FDG)",
"adjacent_complementary_angle(1,CDG,GDB)"
] | {"START": ["angle_addition(1,CDE,EDF)", "angle_addition(1,CDF,FDG)", "adjacent_complementary_angle(1,CDG,GDB)"]} | |
6,858 | DengfengYue_2023-06-04 | GeoQA-angle-10082 | 10 | 如图所示，∠AEO=90°，∠COA=100°，圆O的圆心为O，⊙O的直径为DB。求∠OCD的大小。 | As shown in the diagram, ∠AEO=90°, ∠COA=100°, O is the center of circle O, DB is the diameter of circle O. Find the measure of ∠OCD. | 6858.png | [
"Shape(AO,OD,ODA)",
"Shape(DO,OC,CD)",
"Shape(DC,OCD)",
"Shape(AE,EO,OA)",
"Shape(OE,EC,CO)",
"Shape(BE,EA,OAB)",
"Shape(CE,EB,OBC)",
"Collinear(DOEB)",
"Collinear(AEC)",
"Cocircular(O,DABC)"
] | [
"Equal(MeasureOfAngle(AEO),90)",
"Equal(MeasureOfAngle(COA),100)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(DB,O)"
] | [] | Value(MeasureOfAngle(OCD)) | 25 | [
"arc_property_center_angle(1,OAC,O)",
"circle_property_chord_perpendicular_bisect_arc(1,OAC,OEB)",
"congruent_arc_judgment_length_equal(1,OAB,OBC)",
"congruent_arc_property_measure_equal(1,OAB,OBC)",
"arc_addition_measure(1,OAB,OBC)",
"arc_property_circumference_angle_external(1,OBC,D)",
"radius_of_circ... | {"START": ["arc_property_center_angle(1,OAC,O)", "circle_property_chord_perpendicular_bisect_arc(1,OAC,OEB)", "arc_addition_measure(1,OAB,OBC)", "arc_property_circumference_angle_external(1,OBC,D)", "radius_of_circle_property_length_equal(1,OD,O)", "radius_of_circle_property_length_equal(1,OC,O)"], "circle_property_cho... | |
6,859 | DengfengYue_2023-06-04 | GeoQA-angle-10083 | 6 | 如图所示，∠COA=58°，圆O的圆心为O。求∠BAO的大小。 | As shown in the diagram, ∠COA=58°, O is the center of ⊙O. Find the measure of ∠BAO. | 6859.png | [
"Shape(AB,OBA)",
"Shape(BA,AO,OB)",
"Shape(CO,OA,OAC)",
"Shape(BC,OCB)",
"Collinear(BOC)",
"Cocircular(O,BAC)"
] | [
"Equal(MeasureOfAngle(COA),58)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BAO)) | 29 | [
"arc_property_center_angle(1,OAC,O)",
"arc_property_circumference_angle_external(1,OAC,B)",
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OA,O)",
"isosceles_triangle_judgment_line_equal(1,OBA)",
"isosceles_triangle_property_angle_equal(1,OBA)"
] | {"START": ["arc_property_center_angle(1,OAC,O)", "arc_property_circumference_angle_external(1,OAC,B)", "radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OA,O)"], "isosceles_triangle_judgment_line_equal(1,OBA)": ["isosceles_triangle_property_angle_equal(1,OBA)"], "radius_of_circ... | |
6,860 | DengfengYue_2023-06-04 | GeoQA-angle-10084 | 7 | 如图所示，∠BAC=40°，⊙O的圆心为O。求∠OCB的大小。 | As shown in the diagram, ∠BAC=40°, O is the center of ⊙O. Find the measure of ∠OCB. | 6860.png | [
"Shape(BC,OCB)",
"Shape(BO,OC,CB)",
"Shape(BA,AC,CO,OB)",
"Shape(AB,OBA)",
"Shape(CA,OAC)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(BAC),40)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(OCB)) | 50 | [
"arc_property_circumference_angle_external(1,OCB,A)",
"arc_property_center_angle(1,OCB,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_line_equal(1,OCB)",
"isosceles_triangle_property_angle_equal(1,OCB)",
"triangle_pro... | {"START": ["arc_property_circumference_angle_external(1,OCB,A)", "arc_property_center_angle(1,OCB,O)", "radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OB,O)", "triangle_property_angle_sum(1,BOC)"], "isosceles_triangle_judgment_line_equal(1,OCB)": ["isosceles_triangle_property... | |
6,861 | DengfengYue_2023-06-04 | GeoQA-angle-10085 | 6 | 如图所示，∠BCD=20°，∠DBC=35°，O是圆O的圆心。求∠OAD的大小。 | As shown in the diagram, ∠BCD=20°, ∠DBC=35°, the center of circle O is O. Find the measure of ∠OAD. | 6861.png | [
"Shape(AD,DO,OA)",
"Shape(BD,DA,OAB)",
"Shape(CD,DB,BC)",
"Shape(CB,OBC)",
"Shape(AO,OD,DC,OCA)",
"Collinear(ADC)",
"Collinear(ODB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BCD),20)",
"Equal(MeasureOfAngle(DBC),35)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(OAD)) | 15 | [
"triangle_property_angle_sum(1,ADO)",
"triangle_property_angle_sum(1,CDB)",
"adjacent_complementary_angle(1,CDB,BDA)",
"adjacent_complementary_angle(1,BDA,ADO)",
"arc_property_circumference_angle_external(1,OAB,C)",
"arc_property_center_angle(1,OAB,O)"
] | {"START": ["triangle_property_angle_sum(1,ADO)", "triangle_property_angle_sum(1,CDB)", "adjacent_complementary_angle(1,CDB,BDA)", "adjacent_complementary_angle(1,BDA,ADO)", "arc_property_circumference_angle_external(1,OAB,C)", "arc_property_center_angle(1,OAB,O)"]} | |
6,862 | DengfengYue_2023-06-04 | GeoQA-angle-10086 | 13 | 如图所示，AD=CD，∠FEB=70°，O是圆O的圆心，圆O的直径为AB。求∠ABC的大小。 | As shown in the diagram, AD=CD, ∠FEB=70°, the center of circle O is O, AB is the diameter of circle O. Find the measure of ∠ABC. | 6862.png | [
"Shape(AD,ODA)",
"Shape(DA,AF,FD)",
"Shape(DC,OCD)",
"Shape(DF,FO,OD)",
"Shape(DO,OB,BD)",
"Shape(CD,DB,BC)",
"Shape(CB,OBC)",
"Shape(BE,OEB)",
"Shape(BF,FE,EB)",
"Shape(EF,FA,OAE)",
"Collinear(AFOB)",
"Collinear(DFE)",
"Cocircular(O,DAEBC)"
] | [
"Equal(LengthOfLine(AD),LengthOfLine(CD))",
"Equal(MeasureOfAngle(FEB),70)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(ABC)) | 40 | [
"congruent_arc_judgment_chord_equal(1,ODA,OCD)",
"congruent_arc_property_measure_equal(1,ODA,OCD)",
"arc_property_circumference_angle_external(1,ODA,B)",
"arc_property_circumference_angle_external(1,OBD,A)",
"arc_property_circumference_angle_external(1,OBD,E)",
"arc_property_circumference_angle_external(1... | {"START": ["congruent_arc_judgment_chord_equal(1,ODA,OCD)", "arc_property_circumference_angle_external(1,ODA,B)", "arc_property_circumference_angle_external(1,OBD,A)", "arc_property_circumference_angle_external(1,OBD,E)", "arc_property_circumference_angle_external(1,OCD,B)", "arc_property_center_angle(1,ODA,O)", "radiu... | |
6,863 | DengfengYue_2023-06-04 | GeoQA-angle-10087 | 7 | 如图所示，∠OBA=40°，圆O的圆心为O。求∠ACB的大小。 | As shown in the diagram, ∠OBA=40°, the center of circle O is O. Find the measure of ∠ACB. | 6863.png | [
"Shape(AO,OD,DA)",
"Shape(AD,DB,BA)",
"Shape(AB,OBA)",
"Shape(BD,DC,CB)",
"Shape(BC,OCB)",
"Shape(CD,DO,OA,OAC)",
"Collinear(ADC)",
"Collinear(ODB)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(OBA),40)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(ACB)) | 50 | [
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_line_equal(1,OBA)",
"isosceles_triangle_property_angle_equal(1,OBA)",
"triangle_property_angle_sum(1,AOB)",
"arc_property_center_angle(1,OBA,O)",
"arc_property_circumference_a... | {"START": ["radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "triangle_property_angle_sum(1,AOB)", "arc_property_center_angle(1,OBA,O)", "arc_property_circumference_angle_external(1,OBA,C)"], "isosceles_triangle_judgment_line_equal(1,OBA)": ["isosceles_triangle_property... | |
6,864 | DengfengYue_2023-06-04 | GeoQA-angle-10090 | 4 | 如图所示，∠CDA=120°，⊙O的圆心为O，⊙O的直径为AB。求∠CAB的大小。 | As shown in the diagram, ∠CDA=120°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠CAB. | 6864.png | [
"Shape(AD,ODA)",
"Shape(DC,OCD)",
"Shape(DA,AC,CD)",
"Shape(CA,AB,BC)",
"Shape(CB,OBC)",
"Shape(BA,OAB)",
"Collinear(AOB)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(CDA),120)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CAB)) | 30 | [
"arc_property_circumference_angle_internal(1,OCA,D)",
"arc_property_circumference_angle_external(1,OCA,B)",
"diameter_of_circle_property_right_angle(1,BCA,O)",
"triangle_property_angle_sum(1,CAB)"
] | {"START": ["arc_property_circumference_angle_internal(1,OCA,D)", "arc_property_circumference_angle_external(1,OCA,B)", "diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,CAB)"]} | |
6,865 | DengfengYue_2023-06-04 | GeoQA-angle-10091 | 4 | 如图所示，⌒OEC的角度为92，O是圆O的圆心，CD是⊙O的直径，AO∥DE。求∠OCA的大小。 | As shown in the diagram, the measure of ⌒OEC is 92, O is the center of ⊙O, CD is the diameter of ⊙O, AO is parallel to DE. Find the measure of ∠OCA. | 6865.png | [
"Shape(AC,OCA)",
"Shape(CA,AO,OC)",
"Shape(CO,OE,OEC)",
"Shape(OD,DE,EO)",
"Shape(ED,ODE)",
"Shape(DO,OA,OAD)",
"Collinear(COD)",
"Cocircular(O,CADE)"
] | [
"Equal(MeasureOfArc(OEC),92)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(CD,O)",
"ParallelBetweenLine(AO,DE)"
] | [] | Value(MeasureOfAngle(OCA)) | 23 | [
"arc_property_circumference_angle_external(1,OAD,C)",
"arc_property_circumference_angle_external(1,OEC,D)",
"parallel_property_alternate_interior_angle(2,AO,DE)",
"arc_property_center_angle(1,OAD,O)"
] | {"START": ["arc_property_circumference_angle_external(1,OAD,C)", "arc_property_circumference_angle_external(1,OEC,D)", "parallel_property_alternate_interior_angle(2,AO,DE)", "arc_property_center_angle(1,OAD,O)"]} | |
6,866 | DengfengYue_2023-06-04 | GeoQA-angle-10092 | 7 | 如图所示，∠ABD=32°，⊙O的圆心为O。求∠CAO的大小。 | As shown in the diagram, ∠ABD=32°, O is the center of circle O. Find the measure of ∠CAO. | 6866.png | [
"Shape(BA,OAB)",
"Shape(AB,BD,DA)",
"Shape(AD,DC,CA)",
"Shape(AC,OCA)",
"Shape(CD,DO,OC)",
"Shape(CO,OD,DB,OBC)",
"Collinear(BDC)",
"Collinear(ADO)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(ABD),32)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(CAO)) | 58 | [
"arc_property_circumference_angle_external(1,OCA,B)",
"arc_property_center_angle(1,OCA,O)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"isosceles_triangle_judgment_line_equal(1,OCA)",
"isosceles_triangle_property_angle_equal(1,OCA)",
"triangle_pro... | {"START": ["arc_property_circumference_angle_external(1,OCA,B)", "arc_property_center_angle(1,OCA,O)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OC,O)", "triangle_property_angle_sum(1,CAO)"], "isosceles_triangle_judgment_line_equal(1,OCA)": ["isosceles_triangle_property... | |
6,867 | DengfengYue_2023-06-04 | GeoQA-angle-10093 | 6 | 如图所示，Sub(MeasureOfAngle(DBC)=MeasureOfAngle(OAD))，O是圆O的圆心。求∠BOA的大小。 | As shown in the diagram, Sub(MeasureOfAngle(DBC)=MeasureOfAngle(OAD)), the center of ⊙O is O. Find the measure of ∠BOA. | 6867.png | [
"Shape(AO,OD,DC,OCA)",
"Shape(CD,DB,BC)",
"Shape(CB,OBC)",
"Shape(BD,DA,OAB)",
"Shape(OA,AD,DO)",
"Collinear(ODB)",
"Collinear(ADC)",
"Cocircular(O,ABC)"
] | [
"Equal(Sub(MeasureOfAngle(DBC),MeasureOfAngle(OAD)),40)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BOA)) | 80 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)",
"adjacent_complementary_angle(1,ODC,CDB)",
"adjacent_complementary_angle(1,ADO,ODC)",
"triangle_property_angle_sum(1,CDB)",
"triangle_property_angle_sum(1,OAD)"
] | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,C)", "adjacent_complementary_angle(1,ODC,CDB)", "adjacent_complementary_angle(1,ADO,ODC)", "triangle_property_angle_sum(1,CDB)", "triangle_property_angle_sum(1,OAD)"]} | |
6,868 | DengfengYue_2023-06-04 | GeoQA-angle-10094 | 7 | 如图所示，⌒OAC的长度与弧OEA的长度相等，∠ABC=128°，圆O的圆心为O。求∠CDE的大小。 | As shown in the diagram, the length of ⌒OAC is equal to the length of arc OEA, ∠ABC=128°, the center of ⊙O is O. Find the measure of ∠CDE. | 6868.png | [
"Shape(BA,OAB)",
"Shape(CB,OBC)",
"Shape(DC,OCD)",
"Shape(DE,ODE)",
"Shape(AE,OEA)",
"Shape(AB,BC,CD,DA)",
"Shape(AD,DE,EA)",
"Cocircular(O,ABCDE)"
] | [
"Equal(LengthOfArc(OAC),LengthOfArc(OEA))",
"Equal(MeasureOfAngle(ABC),128)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(CDE)) | 104 | [
"congruent_arc_judgment_length_equal(1,OAC,OEA)",
"congruent_arc_property_measure_equal(1,OAC,OEA)",
"arc_property_circumference_angle_external(1,OCA,B)",
"arc_property_circumference_angle_external(1,OEA,D)",
"arc_property_circumference_angle_internal(1,OAC,B)",
"arc_property_circumference_angle_internal(... | {"START": ["congruent_arc_judgment_length_equal(1,OAC,OEA)", "arc_property_circumference_angle_external(1,OCA,B)", "arc_property_circumference_angle_external(1,OEA,D)", "arc_property_circumference_angle_internal(1,OAC,B)", "arc_property_circumference_angle_internal(1,OCA,D)", "angle_addition(1,CDA,ADE)"], "congruent_ar... | |
6,869 | DengfengYue_2023-06-04 | GeoQA-angle-10095 | 2 | 如图所示，∠BOA=66°，圆O的圆心为O。求∠BCA的大小。 | As shown in the diagram, ∠BOA=66°, O is the center of circle O. Find the measure of ∠BCA. | 6869.png | [
"Shape(BO,OA,OAB)",
"Shape(BC,CA,AO,OB)",
"Shape(AC,OCA)",
"Shape(CB,OBC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BOA),66)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 33 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)"
] | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,C)"]} | |
6,870 | DengfengYue_2023-06-04 | GeoQA-angle-10096 | 2 | 如图所示，∠BCD=140°，O是圆O的圆心。求∠DOB的大小。 | As shown in the diagram, ∠BCD=140°, the center of circle O is O. Find the measure of ∠DOB. | 6870.png | [
"Shape(BA,OAB)",
"Shape(AB,BO,OD,DA)",
"Shape(BC,CD,DO,OB)",
"Shape(CB,OBC)",
"Shape(DC,OCD)",
"Shape(AD,ODA)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(BCD),140)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(DOB)) | 80 | [
"arc_property_circumference_angle_internal(1,OBD,C)",
"arc_property_center_angle(1,OBD,O)"
] | {"START": ["arc_property_circumference_angle_internal(1,OBD,C)", "arc_property_center_angle(1,OBD,O)"]} | |
6,871 | DengfengYue_2023-06-04 | GeoQA-angle-10097 | 2 | 如图所示，∠COB=100°，⊙O的圆心为O。求∠CAB的大小。 | As shown in the diagram, ∠COB=100°, the center of ⊙O is O. Find the measure of ∠CAB. | 6871.png | [
"Shape(BA,OAB)",
"Shape(AB,BO,OC,CA)",
"Shape(CO,OB,OBC)",
"Shape(AC,OCA)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(COB),100)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(CAB)) | 50 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)"]} | |
6,872 | DengfengYue_2023-06-04 | GeoQA-angle-10098 | 4 | 如图所示，∠ADC=75°，O是⊙O的圆心，圆O的直径为AB。求∠CAB的大小。 | As shown in the diagram, ∠ADC=75°, the center of circle O is O, AB is the diameter of ⊙O. Find the measure of ∠CAB. | 6872.png | [
"Shape(AC,OCA)",
"Shape(AE,EC,CA)",
"Shape(CE,EB,BC)",
"Shape(CB,OBC)",
"Shape(BE,ED,ODB)",
"Shape(EA,AD,DE)",
"Shape(DA,OAD)",
"Collinear(AOEB)",
"Collinear(CED)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(ADC),75)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CAB)) | 15 | [
"arc_property_circumference_angle_external(1,OCA,D)",
"arc_property_circumference_angle_external(1,OCA,B)",
"diameter_of_circle_property_right_angle(1,BCA,O)",
"triangle_property_angle_sum(1,CAB)"
] | {"START": ["arc_property_circumference_angle_external(1,OCA,D)", "arc_property_circumference_angle_external(1,OCA,B)", "diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,CAB)"]} | |
6,873 | DengfengYue_2023-06-04 | GeoQA-angle-10099 | 8 | 如图所示，∠DCA=40°，O是圆O的圆心，AB是圆O的直径，⊙O的切线为CD。求∠OBD的大小。 | As shown in the diagram, ∠DCA=40°, O is the center of circle O, AB is the diameter of circle O, the tangent to circle O is CD. Find the measure of ∠OBD. | 6873.png | [
"Shape(DC,CA,ODA)",
"Shape(DB,OBD)",
"Shape(DO,OB,BD)",
"Shape(AO,OD,ODA)",
"Shape(BA,OAB)",
"Collinear(CAOB)",
"Cocircular(O,DAB)"
] | [
"Equal(MeasureOfAngle(DCA),40)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsTangentOfCircle(CD,O)"
] | [] | Value(MeasureOfAngle(OBD)) | 25 | [
"tangent_of_circle_property_perpendicular(1,CD,O,O)",
"triangle_property_angle_sum(1,DOB)",
"triangle_property_angle_sum(1,DCO)",
"adjacent_complementary_angle(1,COD,DOB)",
"radius_of_circle_property_length_equal(1,OD,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_li... | {"START": ["tangent_of_circle_property_perpendicular(1,CD,O,O)", "triangle_property_angle_sum(1,DOB)", "triangle_property_angle_sum(1,DCO)", "adjacent_complementary_angle(1,COD,DOB)", "radius_of_circle_property_length_equal(1,OD,O)", "radius_of_circle_property_length_equal(1,OB,O)"], "isosceles_triangle_judgment_line_e... | |
6,874 | DengfengYue_2023-06-04 | GeoQA-angle-10100 | 6 | 如图所示，DA=DC，∠CBE=45°，⊙O的圆心为O。求∠DAC的大小。 | As shown in the diagram, DA=DC, ∠CBE=45°, O is the center of ⊙O. Find the measure of ∠DAC. | 6874.png | [
"Shape(AD,ODA)",
"Shape(DA,AC,CD)",
"Shape(DC,OCD)",
"Shape(CA,AB,BC)",
"Shape(BA,OAB)",
"Shape(CB,OBC)",
"Collinear(ABE)",
"Cocircular(O,DABC)"
] | [
"Equal(LengthOfLine(DA),LengthOfLine(DC))",
"Equal(MeasureOfAngle(CBE),45)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(DAC)) | 135/2 | [
"adjacent_complementary_angle(1,ABC,CBE)",
"arc_property_circumference_angle_internal(1,OAC,B)",
"arc_property_circumference_angle_external(1,OAC,D)",
"isosceles_triangle_judgment_line_equal(1,DAC)",
"isosceles_triangle_property_angle_equal(1,DAC)",
"triangle_property_angle_sum(1,DAC)"
] | {"START": ["adjacent_complementary_angle(1,ABC,CBE)", "arc_property_circumference_angle_internal(1,OAC,B)", "arc_property_circumference_angle_external(1,OAC,D)", "isosceles_triangle_judgment_line_equal(1,DAC)", "triangle_property_angle_sum(1,DAC)"], "isosceles_triangle_judgment_line_equal(1,DAC)": ["isosceles_triangle_... | |
6,875 | DengfengYue_2023-06-04 | GeoQA-angle-10101 | 2 | 如图所示，∠BCA=36°，⊙O的圆心为O。求∠BOA的大小。 | As shown in the diagram, ∠BCA=36°, O is the center of circle O. Find the measure of ∠BOA. | 6875.png | [
"Shape(AO,OD,DC,OCA)",
"Shape(OA,AD,DO)",
"Shape(BD,DA,OAB)",
"Shape(CD,DB,BC)",
"Shape(CB,OBC)",
"Collinear(ADC)",
"Collinear(ODB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BCA),36)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BOA)) | 72 | [
"arc_property_circumference_angle_external(1,OAB,C)",
"arc_property_center_angle(1,OAB,O)"
] | {"START": ["arc_property_circumference_angle_external(1,OAB,C)", "arc_property_center_angle(1,OAB,O)"]} | |
6,876 | DengfengYue_2023-06-04 | GeoQA-angle-10104 | 3 | 如图所示，AB=AC，∠ABC=70°，圆O的圆心为O。求∠CAB的大小。 | As shown in the diagram, AB=AC, ∠ABC=70°, the center of circle O is O. Find the measure of ∠CAB. | 6876.png | [
"Shape(BA,OAB)",
"Shape(AC,OCA)",
"Shape(AB,BC,CA)",
"Shape(CB,OBC)",
"Cocircular(O,ABC)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(AC))",
"Equal(MeasureOfAngle(ABC),70)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(CAB)) | 40 | [
"isosceles_triangle_judgment_line_equal(1,ABC)",
"isosceles_triangle_property_angle_equal(1,ABC)",
"triangle_property_angle_sum(1,ABC)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,ABC)", "triangle_property_angle_sum(1,ABC)"], "isosceles_triangle_judgment_line_equal(1,ABC)": ["isosceles_triangle_property_angle_equal(1,ABC)"]} | |
6,877 | DengfengYue_2023-06-04 | GeoQA-angle-10106 | 3 | 如图所示，∠CBA=100°，O是圆O的圆心。求∠EDA的大小。 | As shown in the diagram, ∠CBA=100°, the center of circle O is O. Find the measure of ∠EDA. | 6877.png | [
"Shape(AB,OBA)",
"Shape(DA,OAD)",
"Shape(BC,OCB)",
"Shape(AD,DC,CB,BA)",
"Shape(CD,ODC)",
"Collinear(EDC)",
"Cocircular(O,ADCB)"
] | [
"Equal(MeasureOfAngle(CBA),100)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(EDA)) | 100 | [
"arc_property_circumference_angle_internal(1,OCA,B)",
"arc_property_circumference_angle_external(1,OCA,D)",
"adjacent_complementary_angle(1,EDA,ADC)"
] | {"START": ["arc_property_circumference_angle_internal(1,OCA,B)", "arc_property_circumference_angle_external(1,OCA,D)", "adjacent_complementary_angle(1,EDA,ADC)"]} | |
6,878 | DengfengYue_2023-06-04 | GeoQA-angle-10107 | 10 | 如图所示，AD=CD，∠CAB=70°，⊙O的圆心为O，BC是⊙O的直径。求∠BOE的大小。 | As shown in the diagram, AD=CD, ∠CAB=70°, O is the center of ⊙O, BC is the diameter of circle O. Find the measure of ∠BOE. | 6878.png | [
"Shape(BE,OEB)",
"Shape(BF,FE,EB)",
"Shape(EF,FG,GE)",
"Shape(EG,GD,ODE)",
"Shape(DG,GC,CD)",
"Shape(DC,OCD)",
"Shape(CG,GF,FO,OC)",
"Shape(FB,BO,OF)",
"Shape(DA,AE,ODE)",
"Shape(CB,OBC)",
"Collinear(BOC)",
"Collinear(EFO)",
"Collinear(AEB)",
"Collinear(DGFB)",
"Collinear(EGC)",
"Colli... | [
"Equal(LengthOfLine(AD),LengthOfLine(CD))",
"Equal(MeasureOfAngle(CAB),70)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(BC,O)"
] | [] | Value(MeasureOfAngle(BOE)) | 100 | [
"diameter_of_circle_property_right_angle(1,CEB,O)",
"diameter_of_circle_property_right_angle(1,CDB,O)",
"perpendicular_bisector_judgment_per_and_mid(1,BD,CA)",
"perpendicular_bisector_property_distance_equal(1,BD,CA)",
"isosceles_triangle_judgment_line_equal(1,BCA)",
"isosceles_triangle_property_angle_equ... | {"START": ["diameter_of_circle_property_right_angle(1,CEB,O)", "diameter_of_circle_property_right_angle(1,CDB,O)", "triangle_property_angle_sum(1,EBC)", "triangle_property_angle_sum(1,BCA)", "arc_property_circumference_angle_external(1,OEB,C)", "arc_property_center_angle(1,OEB,O)"], "diameter_of_circle_property_right_a... | |
6,879 | DengfengYue_2023-06-04 | GeoQA-angle-10108 | 10 | 如图所示，OB=CD，O是⊙O的圆心，⊙O的直径为BD。求∠BAC的大小。 | As shown in the diagram, OB=CD, O is the center of ⊙O, the diameter of circle O is BD. Find the measure of ∠BAC. | 6879.png | [
"Shape(AB,OBA)",
"Shape(BC,OCB)",
"Shape(BA,AE,EB)",
"Shape(BO,OC,CB)",
"Shape(OE,EC,CO)",
"Shape(CE,ED,DC)",
"Shape(CD,ODC)",
"Shape(DE,EA,OAD)",
"Collinear(BOED)",
"Collinear(AEC)",
"Cocircular(O,BADC)"
] | [
"Equal(LengthOfLine(OB),LengthOfLine(CD))",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(BD,O)"
] | [] | Value(MeasureOfAngle(BAC)) | 60 | [
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OD,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"isosceles_triangle_judgment_line_equal(1,ODC)",
"isosceles_triangle_judgment_line_equal(1,DCO)",
"isosceles_triangle_property_angle_equal(1,ODC)",
"isosc... | {"START": ["radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OD,O)", "radius_of_circle_property_length_equal(1,OC,O)", "triangle_property_angle_sum(1,COD)", "arc_property_circumference_angle_external(1,OCB,A)", "arc_property_circumference_angle_external(1,OCB,D)"], "isosceles_t... | |
6,880 | DengfengYue_2023-06-04 | GeoQA-angle-10109 | 2 | 如图所示，∠DAB=100°，圆O的圆心为O。求∠BOD的大小。 | As shown in the diagram, ∠DAB=100°, the center of ⊙O is O. Find the measure of ∠BOD. | 6880.png | [
"Shape(AD,ODA)",
"Shape(BA,OAB)",
"Shape(AB,BO,OD,DA)",
"Shape(DO,OB,BC,CD)",
"Shape(CB,OBC)",
"Shape(DC,OCD)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(DAB),100)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BOD)) | 160 | [
"arc_property_circumference_angle_internal(1,ODB,A)",
"arc_property_center_angle(1,ODB,O)"
] | {"START": ["arc_property_circumference_angle_internal(1,ODB,A)", "arc_property_center_angle(1,ODB,O)"]} | |
6,881 | DengfengYue_2023-06-04 | GeoQA-angle-10110 | 2 | 如图所示，∠ABD=32°，圆O的圆心为O。求∠AOC的大小。 | As shown in the diagram, ∠ABD=32°, O is the center of circle O. Find the measure of ∠AOC. | 6881.png | [
"Shape(BA,OAB)",
"Shape(AB,BD,DA)",
"Shape(AD,DC,CA)",
"Shape(AC,OCA)",
"Shape(CD,DO,OC)",
"Shape(CO,OD,DB,OBC)",
"Collinear(ADO)",
"Collinear(BDC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(ABD),32)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(AOC)) | 64 | [
"arc_property_circumference_angle_external(1,OCA,B)",
"arc_property_center_angle(1,OCA,O)"
] | {"START": ["arc_property_circumference_angle_external(1,OCA,B)", "arc_property_center_angle(1,OCA,O)"]} | |
6,882 | DengfengYue_2023-06-04 | GeoQA-angle-10111 | 4 | 如图所示，∠CDB=36°，圆O的圆心为O，圆O的直径为AB。求∠ABC的大小。 | As shown in the diagram, ∠CDB=36°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠ABC. | 6882.png | [
"Shape(AC,OCA)",
"Shape(CA,AE,EC)",
"Shape(CE,EB,BC)",
"Shape(CB,OBC)",
"Shape(BE,ED,DB)",
"Shape(BD,ODB)",
"Shape(DE,EA,OAD)",
"Collinear(AOEB)",
"Collinear(CED)",
"Cocircular(O,CADB)"
] | [
"Equal(MeasureOfAngle(CDB),36)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(ABC)) | 54 | [
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_circumference_angle_external(1,OBC,D)",
"diameter_of_circle_property_right_angle(1,BCA,O)",
"triangle_property_angle_sum(1,CAB)"
] | {"START": ["arc_property_circumference_angle_external(1,OBC,A)", "arc_property_circumference_angle_external(1,OBC,D)", "diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,CAB)"]} | |
6,883 | DengfengYue_2023-06-04 | GeoQA-angle-10112 | 6 | 如图所示，∠COA=100°，O是⊙O的圆心，BD是⊙O的直径，AE⊥OE。求∠CDB的大小。 | As shown in the diagram, ∠COA=100°, the center of ⊙O is O, the diameter of ⊙O is BD, AE is perpendicular to OE. Find the measure of ∠CDB. | 6883.png | [
"Shape(AO,OD,ODA)",
"Shape(DO,OC,CD)",
"Shape(DC,OCD)",
"Shape(OA,AE,EO)",
"Shape(OE,EC,CO)",
"Shape(BE,EA,OAB)",
"Shape(CE,EB,OBC)",
"Collinear(DOEB)",
"Collinear(AEC)",
"Cocircular(O,DABC)"
] | [
"Equal(MeasureOfAngle(COA),100)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(BD,O)",
"PerpendicularBetweenLine(AE,OE)"
] | [] | Value(MeasureOfAngle(CDB)) | 25 | [
"arc_property_center_angle(1,OAC,O)",
"circle_property_chord_perpendicular_bisect_arc(1,OAC,OEB)",
"congruent_arc_judgment_length_equal(1,OAB,OBC)",
"congruent_arc_property_measure_equal(1,OAB,OBC)",
"arc_addition_measure(1,OAB,OBC)",
"arc_property_circumference_angle_external(1,OBC,D)"
] | {"START": ["arc_property_center_angle(1,OAC,O)", "circle_property_chord_perpendicular_bisect_arc(1,OAC,OEB)", "arc_addition_measure(1,OAB,OBC)", "arc_property_circumference_angle_external(1,OBC,D)"], "circle_property_chord_perpendicular_bisect_arc(1,OAC,OEB)": ["congruent_arc_judgment_length_equal(1,OAB,OBC)"], "congru... | |
6,884 | DengfengYue_2023-06-04 | GeoQA-angle-10113 | 2 | 如图所示，∠BOA=82°，O是⊙O的圆心。求∠BCA的大小。 | As shown in the diagram, ∠BOA=82°, the center of ⊙O is O. Find the measure of ∠BCA. | 6884.png | [
"Shape(BO,OA,OAB)",
"Shape(AO,OB,BC,CA)",
"Shape(AC,OCA)",
"Shape(CB,OBC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BOA),82)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 41 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)"
] | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,C)"]} | |
6,885 | DengfengYue_2023-06-04 | GeoQA-angle-10114 | 4 | 如图所示，∠CAB=20°，O是⊙O的圆心，⊙O的直径为AB。求∠CDA的大小。 | As shown in the diagram, ∠CAB=20°, the center of circle O is O, AB is the diameter of ⊙O. Find the measure of ∠CDA. | 6885.png | [
"Shape(AD,ODA)",
"Shape(DC,OCD)",
"Shape(CB,OBC)",
"Shape(DA,AC,CD)",
"Shape(CA,AB,BC)",
"Shape(BA,OAB)",
"Collinear(AOB)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(CAB),20)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CDA)) | 110 | [
"diameter_of_circle_property_right_angle(1,BCA,O)",
"triangle_property_angle_sum(1,CAB)",
"arc_property_circumference_angle_internal(1,OAC,B)",
"arc_property_circumference_angle_external(1,OAC,D)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,CAB)", "arc_property_circumference_angle_internal(1,OAC,B)", "arc_property_circumference_angle_external(1,OAC,D)"]} | |
6,886 | ZheZhu_2023-06-04 | GeoQA-angle-10117 | 2 | 如图所示，∠DAB=56°，圆O的圆心为O，⊙O的直径为AB。求∠ABD的大小。 | As shown in the diagram, ∠DAB=56°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠ABD. | 6886.png | [
"Shape(ODA,AD)",
"Shape(DA,AB,BD)",
"Shape(OBD,DB)",
"Collinear(AOB)",
"Cocircular(O,DAB)"
] | [
"Equal(MeasureOfAngle(DAB),56)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(ABD)) | 34 | [
"diameter_of_circle_property_right_angle(1,BDA,O)",
"triangle_property_angle_sum(1,DAB)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BDA,O)", "triangle_property_angle_sum(1,DAB)"]} | |
6,887 | ZheZhu_2023-06-04 | GeoQA-angle-10118 | 4 | 如图所示，∠ACB=67°，∠AOP=42°，⊙O的圆心为O。求∠POB的大小。 | As shown in the diagram, ∠ACB=67°, ∠AOP=42°, O is the center of circle O. Find the measure of ∠POB. | 6887.png | [
"Shape(OB,OBP,PO)",
"Shape(OP,OPA,AO)",
"Shape(OAC,CA)",
"Shape(OCB,BC)",
"Shape(CB,BO,OA,AC)",
"Cocircular(O,CBPA)"
] | [
"Equal(MeasureOfAngle(ACB),67)",
"Equal(MeasureOfAngle(AOP),42)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(POB)) | 92 | [
"arc_property_center_angle(1,OBP,O)",
"arc_property_center_angle(1,OPA,O)",
"arc_property_circumference_angle_external(1,OBA,C)",
"arc_addition_measure(1,OBP,OPA)"
] | {"START": ["arc_property_center_angle(1,OBP,O)", "arc_property_center_angle(1,OPA,O)", "arc_property_circumference_angle_external(1,OBA,C)", "arc_addition_measure(1,OBP,OPA)"]} | |
6,888 | ZheZhu_2023-06-04 | GeoQA-angle-10119 | 3 | 如图所示，∠BAO=40°，∠OHA=90°，圆O的圆心为O。求∠ABC的大小。 | As shown in the diagram, ∠BAO=40°, ∠OHA=90°, O is the center of ⊙O. Find the measure of ∠ABC. | 6888.png | [
"Shape(AO,OH,HA)",
"Shape(AH,HC,OCA)",
"Shape(CH,HB,BC)",
"Shape(OBC,CB)",
"Collinear(OHC)",
"Collinear(AHB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BAO),40)",
"Equal(MeasureOfAngle(OHA),90)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(ABC)) | 25 | [
"triangle_property_angle_sum(1,AOH)",
"arc_property_center_angle(1,OCA,O)",
"arc_property_circumference_angle_external(1,OCA,B)"
] | {"START": ["triangle_property_angle_sum(1,AOH)", "arc_property_center_angle(1,OCA,O)", "arc_property_circumference_angle_external(1,OCA,B)"]} | |
6,889 | ZheZhu_2023-06-04 | GeoQA-angle-10120 | 9 | 如图所示，∠ABC=20°，O是圆O的圆心，AB是⊙O的直径，C平分⌒OBD。求∠DOA的大小。 | As shown in the diagram, ∠ABC=20°, O is the center of circle O, the diameter of ⊙O is AB, C bisects arc OBD. Find the measure of ∠DOA. | 6889.png | [
"Shape(OCA,AF,FC)",
"Shape(OC,CF,FO)",
"Shape(CO,OB,BC)",
"Shape(OBC,CB)",
"Shape(FA,OAD,DF)",
"Shape(OF,FD,DO)",
"Shape(BO,OD,ODB)",
"Collinear(AFOB)",
"Collinear(CFD)",
"Cocircular(O,CADB)"
] | [
"Equal(MeasureOfAngle(ABC),20)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsMidpointOfArc(C,OBD)"
] | [] | Value(MeasureOfAngle(DOA)) | 100 | [
"arc_property_center_angle(1,OCA,O)",
"arc_property_center_angle(1,OCD,O)",
"arc_property_center_angle(1,OBA,O)",
"arc_property_circumference_angle_external(1,OCA,B)",
"arc_addition_measure(1,OBC,OCA)",
"flat_angle(1,AOB)",
"angle_addition(1,DOF,FOC)",
"congruent_arc_judgment_length_equal(1,OCD,OBC)",... | {"START": ["arc_property_center_angle(1,OCA,O)", "arc_property_center_angle(1,OCD,O)", "arc_property_center_angle(1,OBA,O)", "arc_property_circumference_angle_external(1,OCA,B)", "arc_addition_measure(1,OBC,OCA)", "flat_angle(1,AOB)", "angle_addition(1,DOF,FOC)", "congruent_arc_judgment_length_equal(1,OCD,OBC)"], "cong... | |
6,890 | ZheZhu_2023-06-04 | GeoQA-angle-10121 | 8 | 如图所示，∠CAB=50°，∠COE=∠EOB，O是圆O的圆心，E平分线段BC。求∠BDO的大小。 | As shown in the diagram, ∠CAB=50°, ∠COE=∠EOB, the center of ⊙O is O, E bisects segment BC. Find the measure of ∠BDO. | 6890.png | [
"Shape(AB,BO,OC,CA)",
"Shape(OB,BE,EO)",
"Shape(OE,EC,CO)",
"Shape(EB,BD,DE)",
"Shape(OBD,DB)",
"Shape(CE,ED,ODC)",
"Collinear(BEC)",
"Collinear(OED)",
"Cocircular(O,ABDC)"
] | [
"Equal(MeasureOfAngle(CAB),50)",
"Equal(MeasureOfAngle(COE),MeasureOfAngle(EOB))",
"IsCentreOfCircle(O,O)",
"IsMidpointOfLine(E,BC)"
] | [] | Value(MeasureOfAngle(BDO)) | 65 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"angle_addition(1,COD,DOB)",
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OD,O)",
"isosceles_triangle_judgment_line_equal(1,OBD)",
"isosceles_triangle_property_angle... | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "angle_addition(1,COD,DOB)", "radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OD,O)", "triangle_property_angle_sum(1,OBD)"], "isosceles_triangle_judgment_line_equal(1,OBD)": ... | |
6,891 | ZheZhu_2023-06-04 | GeoQA-angle-10122 | 5 | 如图所示，∠CAB=50°，圆O的圆心为O，圆O的直径为AB，AB和AC是等腰三角形ABC的腰。求⌒ODE的角度。 | As shown in the diagram, ∠CAB=50°, the center of circle O is O, AB is the diameter of ⊙O, ∠ABC and ∠BCA are the base angles of the isosceles triangle ABC. Find the measure of ⌒ODE. | 6891.png | [
"Shape(OEA,AE)",
"Shape(EA,AD,ODE)",
"Shape(AO,OB,BD,DA)",
"Shape(ODE,DC,CE)",
"Shape(OBD,DB)",
"Collinear(BDC)",
"Collinear(AOB)",
"Collinear(AEC)",
"Cocircular(O,ABDE)"
] | [
"Equal(MeasureOfAngle(CAB),50)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsoscelesTriangle(ABC)"
] | [] | Value(MeasureOfArc(ODE)) | 50 | [
"diameter_of_circle_property_right_angle(1,BDA,O)",
"altitude_of_triangle_judgment(1,AD,ABC)",
"isosceles_triangle_property_line_coincidence(1,ABC,D)",
"angle_addition(1,CAD,DAB)",
"arc_property_circumference_angle_external(1,ODE,A)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BDA,O)", "angle_addition(1,CAD,DAB)", "arc_property_circumference_angle_external(1,ODE,A)"], "altitude_of_triangle_judgment(1,AD,ABC)": ["isosceles_triangle_property_line_coincidence(1,ABC,D)"], "diameter_of_circle_property_right_angle(1,BDA,O)": ["altitude_of_trian... | |
6,892 | ZheZhu_2023-06-04 | GeoQA-angle-10124 | 19 | 如图所示，CB=CD，∠ACD=45°，∠DAC=30°，O是⊙O的圆心。求∠OAB的大小。 | As shown in the diagram, CB=CD, ∠ACD=45°, ∠DAC=30°, O is the center of circle O. Find the measure of ∠OAB. | 6892.png | [
"Shape(ODA,AD)",
"Shape(OCD,DC)",
"Shape(OBC,CB)",
"Shape(CD,DE,EC)",
"Shape(ED,DA,AE)",
"Shape(EA,AO,OB,BE)",
"Shape(BC,CE,EB)",
"Shape(OA,AB,BO)",
"Collinear(BED)",
"Collinear(AEC)",
"Cocircular(O,ABCD)"
] | [
"Equal(LengthOfLine(CB),LengthOfLine(CD))",
"Equal(MeasureOfAngle(ACD),45)",
"Equal(MeasureOfAngle(DAC),30)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(OAB)) | 15 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,D)",
"arc_property_circumference_angle_external(1,OBD,A)",
"arc_property_circumference_angle_external(1,ODA,B)",
"arc_property_circumference_angle_external(1,OCD,A)",
"arc_property_circumference_angle_external(1,ODA,C)"... | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,D)", "arc_property_circumference_angle_external(1,OBD,A)", "arc_property_circumference_angle_external(1,ODA,B)", "arc_property_circumference_angle_external(1,OCD,A)", "arc_property_circumference_angle_external(1,ODA,C)", "... | |
6,893 | ZheZhu_2023-06-04 | GeoQA-angle-10125 | 1 | 如图所示，∠AOC=Mul(2,MeasureOfAngle(ABC))°，∠COD=Sub(147,Mul(2,MeasureOfAngle(ABC)))°，∠ODC=90°，Sub(MeasureOfAngle(BCA)=MeasureOfAngle(ABC))，O是圆O的圆心。求∠AOD的大小。 | As shown in the diagram, ∠AOC=Mul(2,MeasureOfAngle(ABC))°, ∠COD=Sub(147,Mul(2,MeasureOfAngle(ABC)))°, ∠ODC=90°, Sub(MeasureOfAngle(BCA)=MeasureOfAngle(ABC)), O is the center of ⊙O. Find the measure of ∠AOD. | 6893.png | [
"Shape(AB,BO,OA)",
"Shape(OB,BD,DO)",
"Shape(OD,DC,CO)",
"Shape(AO,OC,CA)",
"Collinear(BDC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(AOC),Mul(2,MeasureOfAngle(ABC)))",
"Equal(MeasureOfAngle(COD),Sub(147,Mul(2,MeasureOfAngle(ABC))))",
"Equal(MeasureOfAngle(ODC),90)",
"Equal(Sub(MeasureOfAngle(BCA),MeasureOfAngle(ABC)),33)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(AOD)) | 147 | [
"angle_addition(1,AOC,COD)"
] | {"START": ["angle_addition(1,AOC,COD)"]} | |
6,894 | ZheZhu_2023-06-04 | GeoQA-angle-10126 | 12 | 如图所示，∠CEO=90°，∠DOC=80°，∠OED=90°，O是圆O的圆心，AB是⊙O的直径。求∠OAC的大小。 | As shown in the diagram, ∠CEO=90°, ∠DOC=80°, ∠OED=90°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠OAC. | 6894.png | [
"Shape(OA,AC,CO)",
"Shape(EO,OC,CE)",
"Shape(DO,OE,ED)",
"Shape(OBD,DE,EB)",
"Shape(OCB,BE,EC)",
"Collinear(AOEB)",
"Collinear(DEC)",
"Cocircular(O,ACBD)"
] | [
"Equal(MeasureOfAngle(CEO),90)",
"Equal(MeasureOfAngle(DOC),80)",
"Equal(MeasureOfAngle(OED),90)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(OAC)) | 20 | [
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"radius_of_circle_property_length_equal(1,OD,O)",
"isosceles_triangle_judgment_line_equal(1,OAC)",
"isosceles_triangle_judgment_line_equal(1,OCD)",
"isosceles_triangle_property_angle_equal(1,OAC)",
"isosc... | {"START": ["radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OD,O)", "triangle_property_angle_sum(1,OAC)", "triangle_property_angle_sum(1,EOC)", "triangle_property_angle_sum(1,OCD)", "flat_angle(1,EOA)", "angle_addition(1,EOC,CO... | |
6,895 | ZheZhu_2023-06-04 | GeoQA-angle-10127 | 4 | 如图所示，∠CAD=25°，O是圆O的圆心，圆O的直径为AB，AD∥OC，DA∥CO。求∠BOC的大小。 | As shown in the diagram, ∠CAD=25°, O is the center of ⊙O, AB is the diameter of circle O, AD∥OC, DA∥CO. Find the measure of ∠BOC. | 6895.png | [
"Shape(CA,AD,ODC)",
"Shape(OA,AC,CO)",
"Shape(BO,OC,OCB)",
"Shape(OAD,DA)",
"Collinear(AOB)",
"Cocircular(O,ADCB)"
] | [
"Equal(MeasureOfAngle(CAD),25)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"ParallelBetweenLine(AD,OC)",
"ParallelBetweenLine(DA,CO)"
] | [] | Value(MeasureOfAngle(BOC)) | 50 | [
"angle_addition(1,OAC,CAD)",
"parallel_property_corresponding_angle(1,OC,AD,B)",
"arc_property_center_angle(1,OCB,O)",
"arc_property_circumference_angle_external(1,OCB,A)"
] | {"START": ["angle_addition(1,OAC,CAD)", "parallel_property_corresponding_angle(1,OC,AD,B)", "arc_property_center_angle(1,OCB,O)", "arc_property_circumference_angle_external(1,OCB,A)"]} | |
6,896 | ZheZhu_2023-06-04 | GeoQA-angle-10128 | 5 | 如图所示，∠CAB=24°，O是⊙O的圆心，AB是圆O的直径。求∠CDA的大小。 | As shown in the diagram, ∠CAB=24°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠CDA. | 6896.png | [
"Shape(DA,AE,ED)",
"Shape(CD,DE,EC)",
"Shape(EA,AO,OB,BE)",
"Shape(CE,EB,OBC)",
"Collinear(AOB)",
"Collinear(DEB)",
"Collinear(AEC)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(CAB),24)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CDA)) | 114 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_circumference_angle_external(1,OAC,D)",
"arc_addition_measure(1,OAB,OBC)",
"flat_angle(1,BOA)"
] | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OBC,A)", "arc_property_circumference_angle_external(1,OAC,D)", "arc_addition_measure(1,OAB,OBC)", "flat_angle(1,BOA)"]} | |
6,897 | ZheZhu_2023-06-04 | GeoQA-angle-10131 | 4 | 如图所示，∠DCE=65°，O是圆O的圆心。求∠DAB的大小。 | As shown in the diagram, ∠DCE=65°, O is the center of circle O. Find the measure of ∠DAB. | 6897.png | [
"Shape(OAB,BA)",
"Shape(OBC,CB)",
"Shape(OCD,DC)",
"Shape(ODA,AD)",
"Shape(DC,CE)",
"Shape(AB,BC,CD,DA)",
"Collinear(BCE)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(DCE),65)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(DAB)) | 65 | [
"arc_property_circumference_angle_external(1,ODB,C)",
"flat_angle(1,BCE)",
"angle_addition(1,BCD,DCE)",
"arc_property_circumference_angle_internal(1,ODB,A)"
] | {"START": ["arc_property_circumference_angle_external(1,ODB,C)", "flat_angle(1,BCE)", "angle_addition(1,BCD,DCE)", "arc_property_circumference_angle_internal(1,ODB,A)"]} | |
6,898 | ZheZhu_2023-06-04 | GeoQA-angle-10132 | 7 | 如图所示，⌒ODA的长度与⌒OCB的长度相等，∠ACE=50°，O是⊙O的圆心。求∠BEC的大小。 | As shown in the diagram, the length of ⌒ODA is equal to the length of ⌒OCB, ∠ACE=50°, the center of circle O is O. Find the measure of ∠BEC. | 6898.png | [
"Shape(EA,AC,CE)",
"Shape(DE,EB,OBD)",
"Shape(EC,OCB,BE)",
"Shape(AE,ED,ODA)",
"Collinear(AEB)",
"Collinear(CED)",
"Cocircular(O,ACBD)"
] | [
"Equal(LengthOfArc(ODA),LengthOfArc(OCB))",
"Equal(MeasureOfAngle(ACE),50)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BEC)) | 100 | [
"congruent_arc_judgment_length_equal(1,ODA,OCB)",
"congruent_arc_property_measure_equal(1,ODA,OCB)",
"arc_property_circumference_angle_external(1,OCB,A)",
"arc_property_circumference_angle_external(1,ODA,C)",
"triangle_property_angle_sum(1,EAC)",
"flat_angle(1,BEA)",
"angle_addition(1,BEC,CEA)"
] | {"START": ["congruent_arc_judgment_length_equal(1,ODA,OCB)", "arc_property_circumference_angle_external(1,OCB,A)", "arc_property_circumference_angle_external(1,ODA,C)", "triangle_property_angle_sum(1,EAC)", "flat_angle(1,BEA)", "angle_addition(1,BEC,CEA)"], "congruent_arc_judgment_length_equal(1,ODA,OCB)": ["congruent_... | |
6,899 | ZheZhu_2023-06-04 | GeoQA-angle-10133 | 8 | 如图所示，AB=AC，∠CAB=50°，⊙O的圆心为O，⊙O的直径为BE。求∠CDB的大小。 | As shown in the diagram, AB=AC, ∠CAB=50°, O is the center of circle O, the diameter of circle O is BE. Find the measure of ∠CDB. | 6899.png | [
"Shape(AB,BO,OD,DA)",
"Shape(DO,OB,BC,CD)",
"Shape(ED,DC,CE)",
"Collinear(ADC)",
"Collinear(EDOB)",
"Cocircular(O,ABCE)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(AC))",
"Equal(MeasureOfAngle(CAB),50)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(BE,O)"
] | [] | Value(MeasureOfAngle(CDB)) | 75 | [
"isosceles_triangle_judgment_line_equal(1,ABC)",
"isosceles_triangle_property_angle_equal(1,ABC)",
"triangle_property_angle_sum(1,DBC)",
"triangle_property_angle_sum(1,ABC)",
"triangle_property_angle_sum(1,BCE)",
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_circumference_angle_ext... | {"START": ["isosceles_triangle_judgment_line_equal(1,ABC)", "triangle_property_angle_sum(1,DBC)", "triangle_property_angle_sum(1,ABC)", "triangle_property_angle_sum(1,BCE)", "arc_property_circumference_angle_external(1,OBC,A)", "arc_property_circumference_angle_external(1,OBC,E)", "diameter_of_circle_property_right_ang... | |
6,900 | ZheZhu_2023-06-04 | GeoQA-angle-10140 | 5 | 如图所示，∠COB=20°，∠OEA=90°，O是圆O的圆心，AB是圆O的直径。求∠ACD的大小。 | As shown in the diagram, ∠COB=20°, ∠OEA=90°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠ACD. | 6900.png | [
"Shape(AE,ED,ODA)",
"Shape(EA,AO,OE)",
"Shape(CD,DE,EC)",
"Shape(CE,EO,OC)",
"Shape(CO,OB,OBC)",
"Shape(OCD,DC)",
"Collinear(AOB)",
"Collinear(DEO)",
"Collinear(AEC)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(COB),20)",
"Equal(MeasureOfAngle(OEA),90)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(ACD)) | 40 | [
"arc_property_center_angle(1,ODA,O)",
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_circumference_angle_external(1,ODA,C)",
"triangle_property_angle_sum(1,EAO)"
] | {"START": ["arc_property_center_angle(1,ODA,O)", "arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "arc_property_circumference_angle_external(1,ODA,C)", "triangle_property_angle_sum(1,EAO)"]} |
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