SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠CDB=65°, AB is parallel to CD, DB is perpendicular to CB. Find the measure of ∠CBA.
问题：如图所示，∠CDB=65°，AB平行于CD，DB垂直于CB。求∠CBA的大小。
Construction (CDL):
Shape(BC,CD,DB)
Shape(CB,BA)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),65)
ParallelBetweenLine(AB,CD)
PerpendicularBetweenLine(DB,CB)
Goal: Value(MeasureOfAngle(CBA))
Answer: 25
Theorem sequence:
triangle_property_angle_sum(1,BCD)
parallel_property_alternate_interior_angle(2,AB,CD)
Problem: As shown in the diagram, ∠DOE=48°, O is the center of ⊙O, OE is perpendicular to DE. Find the measure of ∠OAC.
问题：如图所示，∠DOE=48°，⊙O的圆心为O，OE垂直于DE。求∠OAC的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(ODA,AO,OD)
Shape(OCB,BE,EC)
Shape(OBD,DE,EB)
Shape(AC,CO,OA)
Shape(OC,CE,EO)
Shape(ED,DO,OE)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DOE),48)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,DE)
Goal: Value(MeasureOfAngle(OAC))
Answer: 24
Theorem sequence:
adjacent_complementary_angle(1,CEO,OED)
circle_property_chord_perpendicular_bisect_arc(1,OCD,OEB)
congruent_arc_judgment_length_equal(1,OCB,OBD)
congruent_arc_property_measure_equal(1,OCB,OBD)
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠CAB=40°, ∠CBD=120°. Find the measure of ∠BCA.
问题：如图所示，∠CAB=40°，∠CBD=120°。求∠BCA的大小。
Construction (CDL):
Shape(CA,AB,BC)
Shape(CB,BD)
Collinear(ABD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
Equal(MeasureOfAngle(CBD),120)
Goal: Value(MeasureOfAngle(BCA))
Answer: 80
Theorem sequence:
flat_angle(1,ABD)
angle_addition(1,ABC,CBD)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, AB=AC, ∠CAD=100°, ED is the perpendicular bisector of AB. Find the measure of ∠EAD.
问题：如图所示，AB=AC，∠CAD=100°，ED垂直平分AB。求∠EAD的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BE,ED)
Shape(AE,EC,CA)
Collinear(ADB)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAD),100)
IsPerpendicularBisectorOfLine(ED,AB)
Goal: Value(MeasureOfAngle(EAD))
Answer: 40
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
perpendicular_bisector_property_distance_equal(1,ED,AB)
isosceles_triangle_judgment_line_equal(1,EAB)
isosceles_triangle_property_angle_equal(1,EAB)
Problem: As shown in the diagram, Add(MeasureOfAngle(DAB)=MeasureOfAngle(ABC)), ∠DCF=45°. Find the measure of ∠CDA.
问题：如图所示，Add(MeasureOfAngle(DAB)=MeasureOfAngle(ABC))，∠DCF=45°。求∠CDA的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Shape(EA,AD)
Shape(DC,CF)
Collinear(EAB)
Collinear(BCF)
Conditions (CDL):
Equal(Add(MeasureOfAngle(DAB),MeasureOfAngle(ABC)),180)
Equal(MeasureOfAngle(DCF),45)
Goal: Value(MeasureOfAngle(CDA))
Answer: 45
Theorem sequence:
parallel_judgment_ipsilateral_internal_angle(1,AD,BC)
parallel_property_collinear_extend(1,CB,DA,F)
parallel_property_alternate_interior_angle(2,FC,DA)
Problem: As shown in the diagram, ∠NMA=56°, NH bisects ∠MND, AB is parallel to CD. Find the measure of ∠MNH.
问题：如图所示，∠NMA=56°，NH是∠MND的角平分线，AB平行于CD。求∠MNH的大小。
Construction (CDL):
Shape(AM,ME)
Shape(EM,MB)
Shape(NM,MA)
Shape(BM,MN)
Shape(CN,NM)
Shape(MN,NH)
Shape(HN,ND)
Shape(MN,ND)
Shape(FN,NC)
Shape(DN,NF)
Collinear(AMB)
Collinear(CND)
Collinear(EMNF)
Conditions (CDL):
Equal(MeasureOfAngle(NMA),56)
IsBisectorOfAngle(NH,MND)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(MNH))
Answer: 28
Theorem sequence:
parallel_property_collinear_extend(3,DC,BA,N)
parallel_property_collinear_extend(3,AB,ND,M)
parallel_property_alternate_interior_angle(2,AM,ND)
angle_addition(1,MNH,HND)
Problem: As shown in the diagram, ∠BAD=60°, O is the center of ⊙O. Find the measure of ∠DCB.
问题：如图所示，∠BAD=60°，O是⊙O的圆心。求∠DCB的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(OAD,DA)
Shape(ODC,CD)
Shape(OCB,BC)
Shape(AD,DO,OB,BA)
Shape(OD,DC,CB,BO)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 120
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,A)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,OBD,C)
arc_property_center_angle(1,OBD,O)
round_angle(1,DOB,BOD)
Problem: As shown in the diagram, ∠BOD=140°, the center of ⊙O is O, the diameter of ⊙O is BA. Find the measure of ∠ECA.
问题：如图所示，∠BOD=140°，O是⊙O的圆心，⊙O的直径为BA。求∠ECA的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(OCA,AC)
Shape(BC,CE,EO,OB)
Shape(CA,AE,EC)
Shape(OAD,DE,EA)
Shape(ODB,BO,OD)
Shape(ED,DO,OE)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BOD),140)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BA,O)
Goal: Value(MeasureOfAngle(ECA))
Answer: 20
Theorem sequence:
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
diameter_of_circle_property_right_angle(1,BCA,O)
angle_addition(1,BCE,ECA)
Problem: As shown in the diagram, ∠EAB=50°, CD is parallel to AB. Find the measure of ∠CEF.
问题：如图所示，∠EAB=50°，CD平行于AB。求∠CEF的大小。
Construction (CDL):
Shape(CE,EF)
Shape(FE,ED)
Shape(AE,EC)
Shape(DE,EA)
Shape(EA,AB)
Collinear(CED)
Collinear(AEF)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),50)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(CEF))
Answer: 130
Theorem sequence:
parallel_property_collinear_extend(3,CD,AB,E)
parallel_property_ipsilateral_internal_angle(1,ED,AB)
vertical_angle(1,CEF,DEA)
Problem: As shown in the diagram, ∠DPB=108°, AB is parallel to CD. Find the measure of ∠CQP.
问题：如图所示，∠DPB=108°，AB平行于CD。求∠CQP的大小。
Construction (CDL):
Shape(AP,PD)
Shape(DP,PB)
Shape(QP,PA)
Shape(BP,PQ)
Shape(CQ,QP)
Shape(PQ,QD)
Shape(EQ,QC)
Shape(DQ,QE)
Collinear(APB)
Collinear(CQD)
Collinear(DPQE)
Conditions (CDL):
Equal(MeasureOfAngle(DPB),108)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(CQP))
Answer: 72
Theorem sequence:
flat_angle(1,DPQ)
angle_addition(1,DPB,BPQ)
parallel_property_collinear_extend(3,DC,BA,Q)
parallel_property_collinear_extend(3,AB,CQ,P)
parallel_property_alternate_interior_angle(1,PB,CQ)
Problem: As shown in the diagram, ∠ABC=37°, ∠BCA=26°, CA∥ED. Find the measure of ∠AED.
问题：如图所示，∠ABC=37°，∠BCA=26°，CA∥ED。求∠AED的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(EA,AC)
Shape(AE,ED)
Collinear(BAE)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),37)
Equal(MeasureOfAngle(BCA),26)
ParallelBetweenLine(CA,ED)
Goal: Value(MeasureOfAngle(AED))
Answer: 63
Theorem sequence:
triangle_property_angle_sum(1,ABC)
flat_angle(1,EAB)
angle_addition(1,EAC,CAB)
parallel_property_alternate_interior_angle(2,CA,ED)
Problem: As shown in the diagram, AM=CN, ∠OAD=28°, quadrilateral ADCB is a rhombus. Find the measure of ∠CBO.
问题：如图所示，AM=CN，∠OAD=28°，ADCB是菱形。求∠CBO的大小。
Construction (CDL):
Shape(CB,BO,OC)
Shape(BM,MO,OB)
Shape(MA,AO,OM)
Shape(CO,ON,NC)
Shape(OA,AD,DN,NO)
Collinear(AMB)
Collinear(CND)
Collinear(MON)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AM),LengthOfLine(CN))
Equal(MeasureOfAngle(OAD),28)
Rhombus(ADCB)
Goal: Value(MeasureOfAngle(CBO))
Answer: 62
Theorem sequence:
vertical_angle(1,AOM,CON)
parallel_property_alternate_interior_angle(1,AB,DC)
congruent_triangle_judgment_aas(1,MAO,NCO)
congruent_triangle_property_line_equal(1,MAO,NCO)
mirror_congruent_triangle_judgment_sss(1,CBO,AOB)
mirror_congruent_triangle_property_angle_equal(1,OCB,OBA)
parallel_property_alternate_interior_angle(2,BC,AD)
flat_angle(1,AOC)
angle_addition(1,AOB,BOC)
triangle_property_angle_sum(1,BOC)
Problem: As shown in the diagram, ∠OBC=42°, the center of ⊙O is O, the center of the circumscribed circle of △ABC is O. Find the measure of ∠CAB.
问题：如图所示，∠OBC=42°，⊙O的圆心为O，O是△ABC外接圆的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(CA,AB,BO,OC)
Shape(CO,OB,BC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),42)
IsCentreOfCircle(O,O)
IsCircumcenterOfTriangle(O,ABC)
Goal: Value(MeasureOfAngle(CAB))
Answer: 48
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠EDC=140°, ∠PBA=70°, AB is parallel to DE. Find the measure of ∠PCD.
问题：如图所示，∠EDC=140°，∠PBA=70°，AB平行于DE。求∠PCD的大小。
Construction (CDL):
Shape(DP,PC,CD)
Shape(PB,BA)
Shape(FP,PB)
Shape(BP,PD)
Shape(CP,PF)
Shape(ED,DC)
Collinear(BPC)
Collinear(FPDE)
Conditions (CDL):
Equal(MeasureOfAngle(EDC),140)
Equal(MeasureOfAngle(PBA),70)
ParallelBetweenLine(AB,DE)
Goal: Value(MeasureOfAngle(PCD))
Answer: 30
Theorem sequence:
parallel_property_collinear_extend(2,ED,BA,P)
parallel_property_alternate_interior_angle(2,EP,BA)
flat_angle(1,BPC)
angle_addition(1,BPD,DPC)
flat_angle(1,EDP)
angle_addition(1,EDC,CDP)
triangle_property_angle_sum(1,DPC)
Problem: As shown in the diagram, Add(MeasureOfAngle(EAB)=MeasureOfAngle(CDE)), AE is the angle bisector of ∠DAB, AF is the angle bisector of ∠MAE, DF bisects ∠EDN, AB⊥EB, AE⊥DE. Find the measure of ∠DFA.
问题：如图所示，Add(MeasureOfAngle(EAB)=MeasureOfAngle(CDE))，AE是∠DAB的角平分线，AF平分∠MAE，DF平分∠EDN，AB垂直于EB，AE垂直于DE。求∠DFA的大小。
Construction (CDL):
Shape(MA,AF)
Shape(FD,DN)
Shape(FA,AD,DF)
Shape(AE,ED,DA)
Shape(AB,BE,EA)
Shape(DE,EC,CD)
Collinear(MAB)
Collinear(BEC)
Collinear(NDC)
Conditions (CDL):
Equal(Add(MeasureOfAngle(EAB),MeasureOfAngle(CDE)),90)
IsBisectorOfAngle(AE,DAB)
IsBisectorOfAngle(AF,MAE)
IsBisectorOfAngle(DF,EDN)
PerpendicularBetweenLine(AB,EB)
PerpendicularBetweenLine(AE,DE)
Goal: Value(MeasureOfAngle(DFA))
Answer: 135
Theorem sequence:
adjacent_complementary_angle(1,MAE,EAB)
adjacent_complementary_angle(1,CDE,EDN)
angle_addition(1,MAF,FAE)
angle_addition(1,EDF,FDN)
angle_addition(1,FAD,DAE)
angle_addition(1,EDA,ADF)
triangle_property_angle_sum(1,AED)
triangle_property_angle_sum(1,ADF)
Problem: As shown in the diagram, ∠ABC=70°. Find the measure of ∠CDA.
问题：如图所示，∠ABC=70°。求∠CDA的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BC,CD,DA)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),70)
Goal: Value(MeasureOfAngle(CDA))
Answer: 110
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
Problem: As shown in the diagram, ∠ABI=∠EFB, ∠KCD=110°. Find the measure of ∠FGC.
问题：如图所示，∠ABI=∠EFB，∠KCD=110°。求∠FGC的大小。
Construction (CDL):
Shape(AB,BI)
Shape(IB,BC)
Shape(BC,CK)
Shape(KC,CD)
Shape(DC,CG)
Shape(CG,GH)
Shape(HG,GL)
Shape(LG,GF)
Shape(GF,FJ)
Shape(JF,FE)
Shape(EF,FB)
Shape(FB,BA)
Shape(BF,FG,GC,CB)
Collinear(ABCD)
Collinear(EFGH)
Collinear(IBFJ)
Collinear(KCGL)
Conditions (CDL):
Equal(MeasureOfAngle(ABI),MeasureOfAngle(EFB))
Equal(MeasureOfAngle(KCD),110)
Goal: Value(MeasureOfAngle(FGC))
Answer: 70
Theorem sequence:
parallel_judgment_corresponding_angle(2,FE,BA,I)
parallel_property_collinear_extend(2,AB,EF,C)
parallel_property_collinear_extend(1,FE,CA,G)
vertical_angle(1,KCD,GCB)
parallel_property_ipsilateral_internal_angle(1,GE,CA)
Problem: As shown in the diagram, ∠EBD=52°, O is the center of ⊙O, BA is the diameter of ⊙O. Find the measure of ∠ECB.
问题：如图所示，∠EBD=52°，圆O的圆心为O，圆O的直径为BA。求∠ECB的大小。
Construction (CDL):
Shape(OBD,DB)
Shape(ODA,AD)
Shape(OAC,CE,EO,OA)
Shape(OCB,BC)
Shape(CB,BE,EC)
Shape(EB,BD,DE)
Shape(ED,DA,AO,OE)
Collinear(DEC)
Collinear(AOEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(EBD),52)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BA,O)
Goal: Value(MeasureOfAngle(ECB))
Answer: 38
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, ∠FEH=60°, ∠NHA=50°, AB is parallel to CD, GF⊥EF. Find the measure of ∠FMD.
问题：如图所示，∠FEH=60°，∠NHA=50°，AB平行于CD，GF⊥EF。求∠FMD的大小。
Construction (CDL):
Shape(AH,HN)
Shape(NH,HA)
Shape(CN,NH)
Shape(GN,NC)
Shape(EF,FB)
Shape(BF,FM)
Shape(FM,MD)
Shape(DM,MG)
Shape(EH,HF,FE)
Shape(HN,NM,MF,FH)
Shape(NG,GM,MN)
Collinear(AHFB)
Collinear(CNMD)
Collinear(EHNG)
Collinear(FMG)
Conditions (CDL):
Equal(MeasureOfAngle(FEH),60)
Equal(MeasureOfAngle(NHA),50)
ParallelBetweenLine(AB,CD)
PerpendicularBetweenLine(GF,EF)
Goal: Value(MeasureOfAngle(FMD))
Answer: 20
Theorem sequence:
vertical_angle(1,EHF,NHA)
triangle_property_angle_sum(1,EHF)
angle_addition(1,MFH,HFE)
parallel_property_collinear_extend(3,AB,CD,F)
parallel_property_collinear_extend(3,DC,FA,M)
parallel_property_alternate_interior_angle(2,AF,MD)
Problem: As shown in the diagram, ∠ACB=65°, the center of circle O is O, the tangent to ⊙O is PA, PB is the tangent to ⊙O. Find the measure of ∠BPA.
问题：如图所示，∠ACB=65°，O是圆O的圆心，⊙O的切线为PA，⊙O的切线为PB。求∠BPA的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(OCB,BC)
Shape(AC,CB,BO,OA)
Shape(OBA,AO,OB)
Shape(PA,OBA,BP)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),65)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BPA))
Answer: 50
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
tangent_of_circle_property_perpendicular(1,PB,O,O)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
quadrilateral_property_angle_sum(1,AOBP)
Problem: As shown in the diagram, ∠AME=50°, AB is parallel to CD. Find the measure of ∠MND.
问题：如图所示，∠AME=50°，AB平行于CD。求∠MND的大小。
Construction (CDL):
Shape(AM,ME)
Shape(EM,MB)
Shape(BM,MN)
Shape(MN,ND)
Shape(DN,NF)
Shape(FN,NC)
Shape(CN,NM)
Shape(NM,MA)
Collinear(AMB)
Collinear(CND)
Collinear(EMNF)
Conditions (CDL):
Equal(MeasureOfAngle(AME),50)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(MND))
Answer: 130
Theorem sequence:
vertical_angle(1,AME,BMN)
parallel_property_collinear_extend(3,AB,CD,M)
parallel_property_collinear_extend(3,DC,BM,N)
parallel_property_ipsilateral_internal_angle(1,MB,ND)
Problem: As shown in the diagram, ∠CAB=72°, the center of ⊙O is O. Find the measure of ∠BCO.
问题：如图所示，∠CAB=72°，圆O的圆心为O。求∠BCO的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),72)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 18
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, ∠CAB=36°, BD is the angle bisector of ∠ABC, triangleABC is an isosceles triangle. Find the measure of ∠CDB.
问题：如图所示，∠CAB=36°，BD平分∠ABC，∠ABC和∠BCA是等腰△ABC的底角。求∠CDB的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Collinear(ADC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),36)
IsBisectorOfAngle(BD,ABC)
IsoscelesTriangle(ABC)
Goal: Value(MeasureOfAngle(CDB))
Answer: 72
Theorem sequence:
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
angle_addition(1,ABD,DBC)
triangle_property_angle_sum(1,DBC)
Problem: As shown in the diagram, ∠NFC=55°, MN∥XY, triangle BCA is a right triangle. Find the measure of ∠CBE.
问题：如图所示，∠NFC=55°，MN平行于XY，三角形BCA为直角三角形。求∠CBE的大小。
Construction (CDL):
Shape(MD,DA)
Shape(BD,DM)
Shape(AF,FN)
Shape(NF,FC)
Shape(CE,EY)
Shape(XB,BD)
Shape(AD,DF,FA)
Shape(DB,BC,CF,FD)
Shape(CB,BE,EC)
Collinear(MDFN)
Collinear(XBEY)
Collinear(ADB)
Collinear(AFCE)
Conditions (CDL):
Equal(MeasureOfAngle(NFC),55)
ParallelBetweenLine(MN,XY)
RightTriangle(BCA)
Goal: Value(MeasureOfAngle(CBE))
Answer: 35
Theorem sequence:
parallel_property_collinear_extend(3,MN,XY,F)
parallel_property_collinear_extend(3,YX,NF,E)
parallel_property_alternate_interior_angle(1,FN,XE)
adjacent_complementary_angle(1,ECB,BCA)
triangle_property_angle_sum(1,CBE)
Problem: As shown in the diagram, ∠BAE=46°, ∠ECD=27°, AB is parallel to CD. Find the measure of ∠CEA.
问题：如图所示，∠BAE=46°，∠ECD=27°，AB平行于CD。求∠CEA的大小。
Construction (CDL):
Shape(AE,EB,BA)
Shape(CE,EA)
Shape(EC,CD)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(BAE),46)
Equal(MeasureOfAngle(ECD),27)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(CEA))
Answer: 73
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CD)
triangle_property_angle_sum(1,AEB)
adjacent_complementary_angle(1,CEA,AEB)
Problem: As shown in the diagram, ∠CBE=25°, ∠EDA=43°, AD is parallel to CB. Find the measure of ∠DEB.
问题：如图所示，∠CBE=25°，∠EDA=43°，AD∥CB。求∠DEB的大小。
Construction (CDL):
Shape(ED,DA)
Shape(DE,EB)
Shape(EC,CB,BE)
Collinear(DEC)
Conditions (CDL):
Equal(MeasureOfAngle(CBE),25)
Equal(MeasureOfAngle(EDA),43)
ParallelBetweenLine(AD,CB)
Goal: Value(MeasureOfAngle(DEB))
Answer: 68
Theorem sequence:
parallel_property_alternate_interior_angle(2,AD,CB)
triangle_property_angle_sum(1,ECB)
adjacent_complementary_angle(1,DEB,BEC)
Problem: As shown in the diagram, ∠CDE=30°, ∠EBA=40°, AB∥CD. Find the measure of ∠BED.
问题：如图所示，∠CDE=30°，∠EBA=40°，AB∥CD。求∠BED的大小。
Construction (CDL):
Shape(EB,BA)
Shape(BE,ED)
Shape(EC,CD,DE)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(CDE),30)
Equal(MeasureOfAngle(EBA),40)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BED))
Answer: 70
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CD)
triangle_property_angle_sum(1,ECD)
adjacent_complementary_angle(1,BED,DEC)
Problem: As shown in the diagram, ∠ABC=50°, AC is the angle bisector of ∠FAB, EF∥BC. Find the measure of ∠BCA.
问题：如图所示，∠ABC=50°，AC是∠FAB的角平分线，EF平行于BC。求∠BCA的大小。
Construction (CDL):
Shape(BA,AE)
Shape(FA,AC)
Shape(AB,BC,CA)
Collinear(EAF)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
IsBisectorOfAngle(AC,FAB)
ParallelBetweenLine(EF,BC)
Goal: Value(MeasureOfAngle(BCA))
Answer: 65
Theorem sequence:
angle_addition(1,FAC,CAB)
parallel_property_collinear_extend(3,EF,BC,A)
parallel_property_ipsilateral_internal_angle(1,AF,BC)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠EAF=25°, ∠FEA=40°, AB is parallel to CD. Find the measure of ∠FCD.
问题：如图所示，∠EAF=25°，∠FEA=40°，AB∥CD。求∠FCD的大小。
Construction (CDL):
Shape(AF,FE,EA)
Shape(EF,FB)
Shape(BF,FC)
Shape(CF,FA)
Shape(FC,CD)
Collinear(AFB)
Collinear(EFC)
Conditions (CDL):
Equal(MeasureOfAngle(EAF),25)
Equal(MeasureOfAngle(FEA),40)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(FCD))
Answer: 65
Theorem sequence:
triangle_property_angle_sum(1,EAF)
vertical_angle(1,AFE,BFC)
parallel_property_collinear_extend(3,AB,CD,F)
parallel_property_ipsilateral_internal_angle(1,FB,CD)
Problem: As shown in the diagram, ∠ENB=30°, MG bisects ∠NMD, AB∥CD. Find the measure of ∠NMG.
问题：如图所示，∠ENB=30°，MG平分∠NMD，AB平行于CD。求∠NMG的大小。
Construction (CDL):
Shape(AN,NE)
Shape(EN,NB)
Shape(BN,NM)
Shape(MN,NA)
Shape(CM,MN)
Shape(NM,MG)
Shape(GM,MD)
Shape(DM,MF)
Shape(FM,MC)
Collinear(ANB)
Collinear(CMD)
Collinear(ENMF)
Conditions (CDL):
Equal(MeasureOfAngle(ENB),30)
IsBisectorOfAngle(MG,NMD)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(NMG))
Answer: 15
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,N)
parallel_property_collinear_extend(3,DC,BN,M)
parallel_property_corresponding_angle(1,NB,MD,E)
angle_addition(1,NMG,GMD)
Problem: As shown in the diagram, ∠ABC=40°, ∠ACD=120°. Find the measure of ∠CAB.
问题：如图所示，∠ABC=40°，∠ACD=120°。求∠CAB的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD)
Collinear(BCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),40)
Equal(MeasureOfAngle(ACD),120)
Goal: Value(MeasureOfAngle(CAB))
Answer: 80
Theorem sequence:
adjacent_complementary_angle(1,BCA,ACD)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, AD=8, DB=22, ED=3, DE is parallel to BC. Find the length of line BC.
问题：如图所示，AD=8，DB=22，ED=3，DE平行于BC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),8)
Equal(LengthOfLine(DB),22)
Equal(LengthOfLine(ED),3)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 45/4
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_line_ratio(1,ADE,ABC)
line_addition(1,AD,DB)
Problem: As shown in the diagram, ∠BOA=45°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=45°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OC,OCA,AO)
Shape(CO,OB,BC)
Shape(OA,AB,BO)
Shape(OAB,BA)
Shape(OBC,CB)
Collinear(AOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),45)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 45/2
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, AB=AC, ∠CAB=40°, ED is the perpendicular bisector of AB. Find the measure of ∠EBC.
问题：如图所示，AB=AC，∠CAB=40°，ED垂直平分AB。求∠EBC的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DB,BE)
Shape(EB,BC,CE)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),40)
IsPerpendicularBisectorOfLine(ED,AB)
Goal: Value(MeasureOfAngle(EBC))
Answer: 30
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
perpendicular_bisector_property_distance_equal(1,ED,AB)
isosceles_triangle_judgment_line_equal(1,EAB)
isosceles_triangle_property_angle_equal(1,EAB)
angle_addition(1,ABE,EBC)
Problem: As shown in the diagram, ∠BCE=34°, BC is the angle bisector of ∠EBA, AB is parallel to CD. Find the measure of ∠BED.
问题：如图所示，∠BCE=34°，BC是∠EBA的角平分线，AB∥CD。求∠BED的大小。
Construction (CDL):
Shape(CB,BA)
Shape(BC,CE,EB)
Shape(BE,ED)
Collinear(CED)
Conditions (CDL):
Equal(MeasureOfAngle(BCE),34)
IsBisectorOfAngle(BC,EBA)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BED))
Answer: 68
Theorem sequence:
parallel_property_collinear_extend(3,DC,BA,E)
parallel_property_alternate_interior_angle(2,AB,CE)
angle_addition(1,EBC,CBA)
parallel_property_alternate_interior_angle(2,AB,ED)
Problem: As shown in the diagram, ∠EDF=26°, ∠FED=35°, AB is parallel to CD. Find the measure of ∠EBA.
问题：如图所示，∠EDF=26°，∠FED=35°，AB∥CD。求∠EBA的大小。
Construction (CDL):
Shape(FB,BA)
Shape(CF,FB)
Shape(BF,FD)
Shape(EF,FC)
Shape(FE,ED,DF)
Collinear(BFE)
Collinear(CFD)
Conditions (CDL):
Equal(MeasureOfAngle(EDF),26)
Equal(MeasureOfAngle(FED),35)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(EBA))
Answer: 61
Theorem sequence:
triangle_property_angle_sum(1,FED)
vertical_angle(1,CFB,DFE)
parallel_property_collinear_extend(3,DC,BA,F)
parallel_property_ipsilateral_internal_angle(1,FC,BA)
Problem: As shown in the diagram, ∠HGF=42°, EF is the angle bisector of ∠GEB, CD∥AB. Find the measure of ∠DFE.
问题：如图所示，∠HGF=42°，EF是∠GEB的角平分线，CD∥AB。求∠DFE的大小。
Construction (CDL):
Shape(CG,GH)
Shape(HG,GF)
Shape(GF,FI)
Shape(IF,FD)
Shape(DF,FE)
Shape(FE,EB)
Shape(AE,EG)
Shape(EG,GC)
Shape(GE,EF,FG)
Collinear(CGFD)
Collinear(AEB)
Collinear(HGE)
Collinear(IFE)
Conditions (CDL):
Equal(MeasureOfAngle(HGF),42)
IsBisectorOfAngle(EF,GEB)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(DFE))
Answer: 159
Theorem sequence:
parallel_property_collinear_extend(3,CD,AB,G)
parallel_property_collinear_extend(3,BA,DG,E)
parallel_property_corresponding_angle(1,GD,EB,H)
angle_addition(1,GEF,FEB)
adjacent_complementary_angle(1,HGF,FGE)
triangle_property_angle_sum(1,GEF)
adjacent_complementary_angle(1,DFE,EFG)
Problem: As shown in the diagram, ∠CAB=60°, the center of ⊙O is O. Find the measure of ∠COB.
问题：如图所示，∠CAB=60°，O是⊙O的圆心。求∠COB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OB,OBC,CO)
Shape(OCA,AC)
Shape(AB,BO,OC,CA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠AEF=50°, BA∥DC. Find the measure of ∠EFC.
问题：如图所示，∠AEF=50°，BA平行于DC。求∠EFC的大小。
Construction (CDL):
Shape(AE,EF)
Shape(EF,FC)
Shape(DF,FE)
Shape(FE,EB)
Collinear(AEB)
Collinear(CFD)
Conditions (CDL):
Equal(MeasureOfAngle(AEF),50)
ParallelBetweenLine(BA,DC)
Goal: Value(MeasureOfAngle(EFC))
Answer: 130
Theorem sequence:
parallel_property_collinear_extend(3,BA,DC,E)
parallel_property_collinear_extend(3,CD,AE,F)
parallel_property_ipsilateral_internal_angle(1,EA,FC)