SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠DBA=50°, O is the center of circle O. Find the measure of ∠DCB.
问题：如图所示，∠DBA=50°，O是圆O的圆心。求∠DCB的大小。
Construction (CDL):
Shape(CD,DO,OA,OAC)
Shape(OD,DA,AO)
Shape(CB,BD,DC)
Shape(DB,BA,AD)
Shape(BC,OCB)
Shape(AB,OBA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(DBA),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 40
Theorem sequence:
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)
triangle_property_angle_sum(1,AOB)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠BDE=50°, O is the center of circle O, CE⊥AE. Find the measure of ∠ACE.
问题：如图所示，∠BDE=50°，O是圆O的圆心，CE⊥AE。求∠ACE的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AC,CE,EA)
Shape(AE,ED,ODA)
Shape(BE,EC,OCB)
Shape(EB,BD,DE)
Shape(DB,OBD)
Collinear(AEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BDE),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CE,AE)
Goal: Value(MeasureOfAngle(ACE))
Answer: 40
Theorem sequence:
vertical_angle(1,CEA,DEB)
triangle_property_angle_sum(1,EBD)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, ∠EBF=45°, ∠FCD=65°, DF∥AB, FE is parallel to CA. Find the measure of ∠EFD.
问题：如图所示，∠EBF=45°，∠FCD=65°，DF∥AB，FE平行于CA。求∠EFD的大小。
Construction (CDL):
Shape(EB,BF,FE)
Shape(AE,EF,FD,DA)
Shape(DF,FC,CD)
Collinear(BFC)
Collinear(BEA)
Collinear(ADC)
Conditions (CDL):
Equal(MeasureOfAngle(EBF),45)
Equal(MeasureOfAngle(FCD),65)
ParallelBetweenLine(DF,AB)
ParallelBetweenLine(FE,CA)
Goal: Value(MeasureOfAngle(EFD))
Answer: 70
Theorem sequence:
parallel_property_corresponding_angle(1,FE,CA,B)
parallel_property_corresponding_angle(2,BA,FD,C)
angle_addition(1,BFE,EFD)
adjacent_complementary_angle(1,BFD,DFC)
Problem: As shown in the diagram, the length of ⌒OAC is equal to the length of ⌒ODA, ∠ABD=25°, O is the center of ⊙O. Find the measure of ∠COA.
问题：如图所示，弧OAC的长度与弧ODA的长度相等，∠ABD=25°，O是圆O的圆心。求∠COA的大小。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(AO,OB,BD,ODA)
Shape(BO,OC,OCB)
Shape(DB,OBD)
Collinear(AOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfArc(OAC),LengthOfArc(ODA))
Equal(MeasureOfAngle(ABD),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,B)
congruent_arc_judgment_length_equal(1,ODA,OAC)
congruent_arc_property_measure_equal(1,ODA,OAC)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, ∠BCE=29°, AB is parallel to CD, EC⊥DC. Find the measure of ∠ABF.
问题：如图所示，∠BCE=29°，AB∥CD，EC垂直于DC。求∠ABF的大小。
Construction (CDL):
Shape(AB,BF)
Shape(CB,BA)
Shape(FB,BC,CE)
Shape(EC,CD)
Shape(HC,CB)
Collinear(FBC)
Collinear(HCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCE),29)
ParallelBetweenLine(AB,CD)
PerpendicularBetweenLine(EC,DC)
Goal: Value(MeasureOfAngle(ABF))
Answer: 61
Theorem sequence:
parallel_property_collinear_extend(2,DC,BA,H)
angle_addition(1,BCE,ECD)
adjacent_complementary_angle(1,HCB,BCD)
parallel_property_corresponding_angle(2,CH,BA,F)
Problem: As shown in the diagram, ∠FGE=40°, EF bisects ∠GEA, AB∥CD. Find the measure of ∠EFG.
问题：如图所示，∠FGE=40°，EF是∠GEA的角平分线，AB∥CD。求∠EFG的大小。
Construction (CDL):
Shape(FE,EA)
Shape(CF,FE)
Shape(EF,FG,GE)
Shape(BE,EG)
Shape(EG,GD)
Collinear(AEB)
Collinear(CFGD)
Conditions (CDL):
Equal(MeasureOfAngle(FGE),40)
IsBisectorOfAngle(EF,GEA)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(EFG))
Answer: 70
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,E)
parallel_property_collinear_extend(3,DC,EA,F)
parallel_property_alternate_interior_angle(2,DF,EA)
triangle_property_angle_sum(1,EFG)
Problem: As shown in the diagram, ∠EBI=38°, DE∥FG, AB is perpendicular to CB. Find the measure of ∠CHF.
问题：如图所示，∠EBI=38°，DE∥FG，AB垂直于CB。求∠CHF的大小。
Construction (CDL):
Shape(HB,BD)
Shape(FH,HB)
Shape(BH,HI,IB)
Shape(EB,BI)
Shape(BI,IG)
Shape(CH,HF)
Shape(IH,HC)
Shape(AI,IH)
Shape(GI,IA)
Collinear(DBE)
Collinear(FHIG)
Collinear(BHC)
Collinear(BIA)
Conditions (CDL):
Equal(MeasureOfAngle(EBI),38)
ParallelBetweenLine(DE,FG)
PerpendicularBetweenLine(AB,CB)
Goal: Value(MeasureOfAngle(CHF))
Answer: 52
Theorem sequence:
angle_addition(1,EBI,IBH)
parallel_property_collinear_extend(3,DE,FG,B)
parallel_property_collinear_extend(3,GF,EB,H)
parallel_property_ipsilateral_internal_angle(1,BE,HG)
vertical_angle(1,BHI,CHF)
Problem: As shown in the diagram, ∠CDA=55°, ∠DAC=40°, BE∥FI. Find the measure of ∠FGC.
问题：如图所示，∠CDA=55°，∠DAC=40°，BE平行于FI。求∠FGC的大小。
Construction (CDL):
Shape(BC,CA)
Shape(AC,CD,DA)
Shape(AD,DE)
Shape(FG,GC)
Shape(GC,CB)
Shape(CG,GH,HD,DC)
Shape(ED,DH)
Shape(DH,HI)
Collinear(BCDE)
Collinear(FGHI)
Collinear(ACG)
Collinear(ADH)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),55)
Equal(MeasureOfAngle(DAC),40)
ParallelBetweenLine(BE,FI)
Goal: Value(MeasureOfAngle(FGC))
Answer: 95
Theorem sequence:
triangle_property_angle_sum(1,ACD)
vertical_angle(1,ACD,GCB)
parallel_property_collinear_extend(3,BE,FI,C)
parallel_property_collinear_extend(3,IF,CB,G)
parallel_property_ipsilateral_internal_angle(1,GF,CB)
Problem: As shown in the diagram, AO=PO, ∠OCD=50°, AB∥CD. Find the measure of ∠PAO.
问题：如图所示，AO=PO，∠OCD=50°，AB∥CD。求∠PAO的大小。
Construction (CDL):
Shape(AO,OP,PA)
Shape(PO,OB)
Shape(CO,OA)
Shape(BO,OC)
Shape(OC,CD)
Collinear(AOB)
Collinear(POC)
Conditions (CDL):
Equal(LengthOfLine(AO),LengthOfLine(PO))
Equal(MeasureOfAngle(OCD),50)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(PAO))
Answer: 25
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,O)
parallel_property_ipsilateral_internal_angle(1,OB,CD)
vertical_angle(1,AOP,BOC)
isosceles_triangle_judgment_line_equal(1,OPA)
isosceles_triangle_property_angle_equal(1,OPA)
triangle_property_angle_sum(1,AOP)
Problem: As shown in the diagram, MN=100, MO=60, NO=80, ∠MOE=20°. Find the measure of ∠FON.
问题：如图所示，MN=100，MO=60，NO=80，∠MOE=20°。求∠FON的大小。
Construction (CDL):
Shape(NO,OM,MN)
Shape(MO,OE)
Shape(FO,ON)
Collinear(EOF)
Conditions (CDL):
Equal(LengthOfLine(MN),100)
Equal(LengthOfLine(MO),60)
Equal(LengthOfLine(NO),80)
Equal(MeasureOfAngle(MOE),20)
Goal: Value(MeasureOfAngle(FON))
Answer: 70
Theorem sequence:
right_triangle_judgment_pythagorean_inverse(1,NOM)
angle_addition(1,NOM,MOE)
adjacent_complementary_angle(1,FON,NOE)
Problem: As shown in the diagram, ∠AOE=36°, EO⊥DO. Find the measure of ∠DOB.
问题：如图所示，∠AOE=36°，EO⊥DO。求∠DOB的大小。
Construction (CDL):
Shape(CO,OA)
Shape(AO,OE)
Shape(EO,OD)
Shape(DO,OB)
Collinear(AOB)
Collinear(COD)
Conditions (CDL):
Equal(MeasureOfAngle(AOE),36)
PerpendicularBetweenLine(EO,DO)
Goal: Value(MeasureOfAngle(DOB))
Answer: 54
Theorem sequence:
angle_addition(1,AOE,EOD)
adjacent_complementary_angle(1,AOD,DOB)
Problem: As shown in the diagram, ∠CAB=30°, O is the center of ⊙O, the tangent to ⊙O is AB. Find the measure of ∠BDC.
问题：如图所示，∠CAB=30°，O是⊙O的圆心，圆O的切线为AB。求∠BDC的大小。
Construction (CDL):
Shape(CD,ODC)
Shape(OB,BD,DC,CO)
Shape(DB,OBD)
Shape(CA,AB,OCB)
Shape(BO,OC,OCB)
Collinear(ACO)
Cocircular(O,CBD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 30
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AB,O,O)
triangle_property_angle_sum(1,ABO)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, ∠ABD=24°, CB and CA are the legs of the isosceles right △ CBA, DB is parallel to EC. Find the measure of ∠ECA.
问题：如图所示，∠ABD=24°，CB和CA是等腰直角三角形CBA的腰，DB∥EC。求∠ECA的大小。
Construction (CDL):
Shape(AC,CB,BA)
Shape(AB,BD)
Shape(EC,CA)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),24)
IsoscelesRightTriangle(CBA)
ParallelBetweenLine(DB,EC)
Goal: Value(MeasureOfAngle(ECA))
Answer: 21
Theorem sequence:
angle_addition(1,CBA,ABD)
angle_addition(1,ECA,ACB)
isosceles_triangle_property_angle_equal(1,CBA)
triangle_property_angle_sum(1,CBA)
parallel_property_ipsilateral_internal_angle(1,CE,BD)
Problem: As shown in the diagram, AC=5, BC=3, AB is perpendicular to CB. Find the sum of the length of line AB and the length of line BC.
问题：如图所示，AC=5，BC=3，AB垂直于CB。求直线AB的长度与直线BC的长度之和。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AC),5)
Equal(LengthOfLine(BC),3)
PerpendicularBetweenLine(AB,CB)
Goal: Value(Add(LengthOfLine(AB),LengthOfLine(BC)))
Answer: 7
Theorem sequence:
right_triangle_judgment_angle(1,ABC)
right_triangle_property_pythagorean(1,ABC)
Problem: As shown in the diagram, AB=AC, ∠DAC=70°, AD∥BC. Find the measure of ∠CAB.
问题：如图所示，AB=AC，∠DAC=70°，AD平行于BC。求∠CAB的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DA,AC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(DAC),70)
ParallelBetweenLine(AD,BC)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,BC)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, △ABC is congruent to △FEC, ∠ABC=110°, ∠BCE=60°, ∠CAB=40°. Find the measure of ∠BCF.
问题：如图所示，三角形ABC与三角形FEC是全等三角形，∠ABC=110°，∠BCE=60°，∠CAB=40°。求∠BCF的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CE)
Shape(EC,CF,FE)
Conditions (CDL):
CongruentBetweenTriangle(ABC,FEC)
Equal(MeasureOfAngle(ABC),110)
Equal(MeasureOfAngle(BCE),60)
Equal(MeasureOfAngle(CAB),40)
Goal: Value(MeasureOfAngle(BCF))
Answer: 90
Theorem sequence:
congruent_triangle_property_angle_equal(1,ABC,FEC)
congruent_triangle_property_angle_equal(1,BCA,ECF)
triangle_property_angle_sum(1,ECF)
angle_addition(1,BCE,ECF)
Problem: As shown in the diagram, △ABC is congruent to △EDC, ∠DCO=40°, ∠ECA=40°. Find the sum of the measure of ∠OBD and the measure of ∠OEB.
问题：如图所示，三角形ABC与三角形EDC是全等三角形，∠DCO=40°，∠ECA=40°。求∠OBD的大小与∠OEB的大小之和。
Construction (CDL):
Shape(BD,DO,OB)
Shape(BO,OE,EB)
Shape(OD,DC,CO)
Shape(EO,OC,CE)
Shape(AE,EC,CA)
Collinear(BOC)
Collinear(DOE)
Collinear(BEA)
Conditions (CDL):
CongruentBetweenTriangle(ABC,EDC)
Equal(MeasureOfAngle(DCO),40)
Equal(MeasureOfAngle(ECA),40)
Goal: Value(Add(MeasureOfAngle(OBD),MeasureOfAngle(OEB)))
Answer: 110
Theorem sequence:
congruent_triangle_property_line_equal(1,ABC,EDC)
isosceles_triangle_judgment_line_equal(1,CBD)
isosceles_triangle_property_angle_equal(1,CBD)
triangle_property_angle_sum(1,CBD)
congruent_triangle_property_angle_equal(1,ABC,EDC)
triangle_property_angle_sum(1,CAE)
angle_addition(1,AEC,CEO)
adjacent_complementary_angle(1,AEO,OEB)
Problem: As shown in the diagram, ∠EOA=40°, the center of ⊙O is O, AE is perpendicular to OE. Find the measure of ∠BDE.
问题：如图所示，∠EOA=40°，O是圆O的圆心，AE垂直于OE。求∠BDE的大小。
Construction (CDL):
Shape(CE,EA,OAC)
Shape(AE,EO,OA)
Shape(AO,OD,ODA)
Shape(BE,EC,OCB)
Shape(DO,OB,BD)
Shape(OE,EB,BO)
Shape(DB,OBD)
Collinear(CEOD)
Collinear(AEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(EOA),40)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AE,OE)
Goal: Value(MeasureOfAngle(BDE))
Answer: 20
Theorem sequence:
adjacent_complementary_angle(1,AEO,OEB)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
mirror_congruent_triangle_judgment_hl(1,AEO,BOE)
mirror_congruent_triangle_property_angle_equal(1,OAE,OEB)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, ∠CDB=50°, BC is the angle bisector of ∠DBA, AB is parallel to CD. Find the measure of ∠CBA.
问题：如图所示，∠CDB=50°，BC是∠DBA的角平分线，AB∥CD。求∠CBA的大小。
Construction (CDL):
Shape(AB,BE)
Shape(CB,BA)
Shape(BC,CD,DB)
Collinear(EBD)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),50)
IsBisectorOfAngle(BC,DBA)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(CBA))
Answer: 65
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CD)
triangle_property_angle_sum(1,BCD)
Problem: As shown in the diagram, △ABC is congruent to △DEC, ∠CAB=70°. Find the measure of ∠ACD.
问题：如图所示，△ABC全等于△DEC，∠CAB=70°。求∠ACD的大小。
Construction (CDL):
Shape(EO,OA,AE)
Shape(OB,BC,CO)
Shape(AO,OC,CA)
Shape(AC,CD,DA)
Collinear(EOC)
Collinear(BOA)
Collinear(EAD)
Conditions (CDL):
CongruentBetweenTriangle(ABC,DEC)
Equal(MeasureOfAngle(CAB),70)
Goal: Value(MeasureOfAngle(ACD))
Answer: 40
Theorem sequence:
congruent_triangle_property_angle_equal(1,ABC,DEC)
congruent_triangle_property_line_equal(1,BCA,ECD)
isosceles_triangle_judgment_line_equal(1,CDA)
isosceles_triangle_property_angle_equal(1,CDA)
triangle_property_angle_sum(1,CDA)
Problem: As shown in the diagram, ∠DAB=36°, the center of circle O is O. Find the measure of ∠COB.
问题：如图所示，∠DAB=36°，⊙O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(AB,CD,DA)
Shape(BC,CD,DB)
Shape(DC,CO,OD)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AD,DO,OC,OCA)
Collinear(ADC)
Collinear(BDO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),36)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 72
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, △ABC is congruent to △EDC, ∠ABC=50°, ∠BCA=90°. Find the measure of ∠EOC.
问题：如图所示，△ABC全等于△EDC，∠ABC=50°，∠BCA=90°。求∠EOC的大小。
Construction (CDL):
Shape(AD,DO,OA)
Shape(DB,BC,CD)
Shape(DC,CO,OD)
Shape(OC,CE,EO)
Collinear(ADB)
Collinear(DOE)
Collinear(AOC)
Conditions (CDL):
CongruentBetweenTriangle(ABC,EDC)
Equal(MeasureOfAngle(ABC),50)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(MeasureOfAngle(EOC))
Answer: 60
Theorem sequence:
triangle_property_angle_sum(1,ABC)
congruent_triangle_property_line_equal(1,ABC,EDC)
isosceles_triangle_judgment_line_equal(1,CDB)
congruent_triangle_property_angle_equal(1,BCA,DCE)
isosceles_triangle_property_angle_equal(1,CDB)
angle_addition(1,ODC,CDB)
adjacent_complementary_angle(1,ADO,ODB)
triangle_property_angle_sum(1,ADO)
adjacent_complementary_angle(1,COD,DOA)
adjacent_complementary_angle(1,EOC,COD)
Problem: As shown in the diagram, ∠CDA=110°, CF∥GJ, KA is parallel to LB. Find the measure of ∠EIJ.
问题：如图所示，∠CDA=110°，CF平行于GJ，KA∥LB。求∠EIJ的大小。
Construction (CDL):
Shape(CD,DA)
Shape(AD,DE)
Shape(DE,EB)
Shape(BE,EF)
Shape(HD,DC)
Shape(FE,EI)
Shape(GH,HD)
Shape(EI,IJ)
Shape(KH,HG)
Shape(IH,HK)
Shape(LI,IH)
Shape(JI,IL)
Shape(DH,HI,IE,ED)
Collinear(CDEF)
Collinear(GHIJ)
Collinear(ADHK)
Collinear(BEIL)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),110)
ParallelBetweenLine(CF,GJ)
ParallelBetweenLine(KA,LB)
Goal: Value(MeasureOfAngle(EIJ))
Answer: 70
Theorem sequence:
parallel_property_collinear_extend(3,KA,LB,D)
parallel_property_collinear_extend(3,BL,AD,E)
parallel_property_corresponding_angle(1,DA,EB,C)
adjacent_complementary_angle(1,DEB,BEF)
parallel_property_collinear_extend(3,CF,GJ,E)
parallel_property_collinear_extend(3,JG,FE,I)
parallel_property_corresponding_angle(1,EF,IJ,B)
Problem: As shown in the diagram, ∠CAE=30°, O is the center of ⊙O, DC is the diameter of ⊙O. Find the measure of ∠BCE.
问题：如图所示，∠CAE=30°，O是圆O的圆心，DC是⊙O的直径。求∠BCE的大小。
Construction (CDL):
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(AC,OCA)
Shape(CB,OBC)
Shape(DO,OE,EA,AD)
Shape(DA,OAD)
Shape(BE,EO,OD,DB)
Shape(BD,ODB)
Collinear(AEB)
Collinear(CEOD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAE),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DC,O)
Goal: Value(MeasureOfAngle(BCE))
Answer: 60
Theorem sequence:
diameter_of_circle_property_right_angle(1,CAD,O)
angle_addition(1,CAE,EAD)
arc_property_circumference_angle_external(1,ODB,A)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠BOA=60°, O is the center of ⊙O, AB is the tangent to circle O. Find the measure of ∠CAB.
问题：如图所示，∠BOA=60°，O是圆O的圆心，⊙O的切线为AB。求∠CAB的大小。
Construction (CDL):
Shape(BO,OC,OCB)
Shape(CA,AB,OCB)
Shape(CO,OB,OBC)
Collinear(ACO)
Cocircular(O,CB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),60)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 30
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AB,O,O)
triangle_property_angle_sum(1,ABO)
Problem: As shown in the diagram, ∠BGE=140°, ∠FEG=∠DEF, AD is parallel to BC. Find the measure of ∠EFC.
问题：如图所示，∠BGE=140°，∠FEG=∠DEF，AD∥BC。求∠EFC的大小。
Construction (CDL):
Shape(AB,BG,GE,EA)
Shape(EG,GF,FE)
Shape(GH,HI,IF,FG)
Shape(EF,FC,CD,DE)
Collinear(AED)
Collinear(BGFC)
Collinear(EGH)
Conditions (CDL):
Equal(MeasureOfAngle(BGE),140)
Equal(MeasureOfAngle(FEG),MeasureOfAngle(DEF))
ParallelBetweenLine(AD,BC)
Goal: Value(MeasureOfAngle(EFC))
Answer: 110
Theorem sequence:
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_collinear_extend(3,CB,EA,G)
parallel_property_ipsilateral_internal_angle(1,GB,EA)
adjacent_complementary_angle(1,DEG,GEA)
angle_addition(1,DEF,FEG)
parallel_property_collinear_extend(3,CB,DE,F)
parallel_property_ipsilateral_internal_angle(1,ED,FC)
Problem: As shown in the diagram, AD=DiameterOfCircle(O), ∠CBA=35°, the center of circle O is O, DA is the diameter of circle O. Find the measure of ∠DAC.
问题：如图所示，AD=DiameterOfCircle(O)，∠CBA=35°，圆O的圆心为O，DA是⊙O的直径。求∠DAC的大小。
Construction (CDL):
Shape(DO,OE,EC,CD)
Shape(CE,EA,AC)
Shape(EB,BA,AE)
Shape(DC,OCD)
Shape(BE,EO,OD,ODB)
Shape(AB,OBA)
Shape(CA,OAC)
Collinear(DOEA)
Collinear(BEC)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(LengthOfLine(AD),DiameterOfCircle(O))
Equal(MeasureOfAngle(CBA),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DA,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
diameter_of_circle_property_right_angle(1,ACD,O)
triangle_property_angle_sum(1,ACD)
Problem: As shown in the diagram, ∠ACD=50°, BD is parallel to EG. Find the measure of ∠HFE.
问题：如图所示，∠ACD=50°，BD平行于EG。求∠HFE的大小。
Construction (CDL):
Shape(BC,CA)
Shape(AC,CD)
Shape(FC,CB)
Shape(DC,CF)
Shape(EF,FC)
Shape(CF,FG)
Shape(HF,FE)
Shape(GF,FH)
Collinear(BCD)
Collinear(EFG)
Collinear(ACFH)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),50)
ParallelBetweenLine(BD,EG)
Goal: Value(MeasureOfAngle(HFE))
Answer: 50
Theorem sequence:
vertical_angle(1,ACD,FCB)
parallel_property_collinear_extend(3,BD,EG,C)
parallel_property_collinear_extend(3,GE,CB,F)
parallel_property_corresponding_angle(1,FE,CB,H)
Problem: As shown in the diagram, ∠QMA=50°, ∠TRQ=110°, AB∥CD. Find the measure of ∠PEM.
问题：如图所示，∠QMA=50°，∠TRQ=110°，AB∥CD。求∠PEM的大小。
Construction (CDL):
Shape(EM,MP,PE)
Shape(AM,ME)
Shape(EP,PB)
Shape(QM,MA)
Shape(BP,PR)
Shape(MQ,QR,RP,PM)
Shape(CQ,QM)
Shape(PR,RD)
Shape(SQ,QC)
Shape(RQ,QS)
Shape(TR,RQ)
Shape(DR,RT)
Collinear(AMPB)
Collinear(CQRD)
Collinear(EMQS)
Collinear(EPRT)
Conditions (CDL):
Equal(MeasureOfAngle(QMA),50)
Equal(MeasureOfAngle(TRQ),110)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(PEM))
Answer: 60
Theorem sequence:
vertical_angle(1,QMA,EMP)
parallel_property_collinear_extend(3,AB,CD,P)
parallel_property_collinear_extend(3,DC,PA,R)
parallel_property_corresponding_angle(1,RC,PA,T)
adjacent_complementary_angle(1,RPM,MPE)
triangle_property_angle_sum(1,EMP)
Problem: As shown in the diagram, ∠BCA is the right angle of the isosceles right triangle CAB, DF∥PA, PA∥GH. Find the sum of the measure of ∠FEA and the measure of ∠ACH.
问题：如图所示，CA和CB是等腰直角△CAB的腰，DF平行于PA，PA平行于GH。求∠FEA的大小与∠ACH的大小之和。
Construction (CDL):
Shape(BO,OE,EB)
Shape(DO,OB)
Shape(BE,EF)
Shape(CO,OD)
Shape(FE,EA)
Shape(OP,PA,AE,EO)
Shape(PC,CA,AP)
Shape(GC,CO)
Shape(AC,CH)
Collinear(DOEF)
Collinear(GCH)
Collinear(BOPC)
Collinear(BEA)
Conditions (CDL):
IsoscelesRightTriangle(CAB)
ParallelBetweenLine(DF,PA)
ParallelBetweenLine(PA,GH)
Goal: Value(Add(MeasureOfAngle(FEA),MeasureOfAngle(ACH)))
Answer: 45
Theorem sequence:
parallel_property_collinear_extend(3,DF,PA,E)
parallel_property_alternate_interior_angle(1,EF,PA)
parallel_property_collinear_extend(3,HG,AP,C)
parallel_property_alternate_interior_angle(2,PA,CH)
angle_addition(1,CAP,PAE)
isosceles_triangle_property_angle_equal(1,CAB)
triangle_property_angle_sum(1,BCA)
Problem: As shown in the diagram, AB=DiameterOfCircle(O), ∠ACD=35°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠DAB.
问题：如图所示，AB=DiameterOfCircle(O)，∠ACD=35°，O是圆O的圆心，AB是⊙O的直径。求∠DAB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(CA,OAC)
Shape(DA,AE,ED)
Shape(AC,CE,EA)
Shape(DE,EO,OB,BD)
Shape(DB,OBD)
Shape(BO,OE,EC,OCB)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfLine(AB),DiameterOfCircle(O))
Equal(MeasureOfAngle(ACD),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,BDA)
Problem: As shown in the diagram, AB=DiameterOfCircle(O), ∠CAB=23°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠ADC.
问题：如图所示，AB=DiameterOfCircle(O)，∠CAB=23°，⊙O的圆心为O，AB是⊙O的直径。求∠ADC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(DA,OAD)
Shape(BE,ED,ODB)
Shape(CB,OBC)
Shape(BC,CE,EB)
Shape(CA,AO,OE,EC)
Shape(EO,OA,AD,DE)
Collinear(CED)
Collinear(BEOA)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(AB),DiameterOfCircle(O))
Equal(MeasureOfAngle(CAB),23)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 67
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, AB=DiameterOfCircle(O), ∠CAB=22°, O is the center of circle O, the diameter of ⊙O is AB. Find the measure of ∠ADC.
问题：如图所示，AB=DiameterOfCircle(O)，∠CAB=22°，⊙O的圆心为O，⊙O的直径为AB。求∠ADC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(DA,OAD)
Shape(BE,ED,ODB)
Shape(CB,OBC)
Shape(BC,CE,EB)
Shape(CA,AO,OE,EC)
Shape(EO,OA,AD,DE)
Collinear(CED)
Collinear(BEOA)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(AB),DiameterOfCircle(O))
Equal(MeasureOfAngle(CAB),22)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 68
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠ADC=27°, O is the center of ⊙O, BA is the tangent to circle O. Find the measure of ∠CBA.
问题：如图所示，∠ADC=27°，圆O的圆心为O，BA是⊙O的切线。求∠CBA的大小。
Construction (CDL):
Shape(OC,CE,EO)
Shape(CB,BA,OCA)
Shape(AE,EC,OCA)
Shape(EA,AD,DE)
Shape(DA,OAD)
Shape(CO,OE,ED,ODC)
Collinear(CED)
Collinear(OCB)
Collinear(OEA)
Cocircular(O,ADC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),27)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BA,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 36
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_center_angle(1,OCA,O)
tangent_of_circle_property_perpendicular(2,BA,O,O)
triangle_property_angle_sum(1,OBA)
Problem: As shown in the diagram, ∠BEF=125°, ∠CDA=100°, ∠GHD=100°, CF∥GJ. Find the measure of ∠HIE.
问题：如图所示，∠BEF=125°，∠CDA=100°，∠GHD=100°，CF平行于GJ。求∠HIE的大小。
Construction (CDL):
Shape(CD,DA)
Shape(AD,DE)
Shape(DE,EB)
Shape(BE,EF)
Shape(HD,DC)
Shape(FE,EI)
Shape(GH,HD)
Shape(EI,IJ)
Shape(KH,HG)
Shape(IH,HK)
Shape(LI,IH)
Shape(JI,IL)
Shape(DH,HI,IE,ED)
Collinear(CDEF)
Collinear(GHIJ)
Collinear(ADHK)
Collinear(BEIL)
Conditions (CDL):
Equal(MeasureOfAngle(BEF),125)
Equal(MeasureOfAngle(CDA),100)
Equal(MeasureOfAngle(GHD),100)
ParallelBetweenLine(CF,GJ)
Goal: Value(MeasureOfAngle(HIE))
Answer: 55
Theorem sequence:
vertical_angle(1,CDA,EDH)
vertical_angle(1,BEF,IED)
adjacent_complementary_angle(1,GHD,DHI)
quadrilateral_property_angle_sum(1,DHIE)
Problem: As shown in the diagram, Add(MeasureOfAngle(ABC)=MeasureOfAngle(CDA)), AB=DC, AD=BC. Find the measure of ∠DAB.
问题：如图所示，Add(MeasureOfAngle(ABC)=MeasureOfAngle(CDA))，AB=DC，AD=BC。求∠DAB的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Conditions (CDL):
Equal(Add(MeasureOfAngle(ABC),MeasureOfAngle(CDA)),120)
Equal(LengthOfLine(AB),LengthOfLine(DC))
Equal(LengthOfLine(AD),LengthOfLine(BC))
Goal: Value(MeasureOfAngle(DAB))
Answer: 120
Theorem sequence:
parallelogram_judgment_equal_and_equal(1,ABCD)
quadrilateral_property_angle_sum(1,ABCD)
parallelogram_property_opposite_angle_equal(1,ABCD)