SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠CED=30°, ∠DAB=55°, the center of ⊙O is O. Find the measure of ∠BFC.
问题：如图所示，∠CED=30°，∠DAB=55°，O是圆O的圆心。求∠BFC的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(OCD,DC)
Shape(OBC,CB)
Shape(DA,AB,BC,CD)
Shape(OCD,CE,ED)
Shape(OBC,BF,FC)
Shape(O)
Collinear(ADE)
Collinear(ABF)
Collinear(DCF)
Collinear(ECB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CED),30)
Equal(MeasureOfAngle(DAB),55)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BFC))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,ABE)
flat_angle(1,ABF)
angle_addition(1,ABC,CBF)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_internal(1,OBD,C)
flat_angle(1,FCD)
angle_addition(1,FCB,BCD)
triangle_property_angle_sum(1,FCB)
Problem: As shown in the diagram, triangle AOB is congruent to triangle DOC, ∠AOD=30°. Find the measure of ∠ODA.
问题：如图所示，△AOB全等于△DOC，∠AOD=30°。求∠ODA的大小。
Construction (CDL):
Shape(DA,AO,OD)
Shape(ED,DO,OE)
Shape(BD,DE,EB)
Shape(EO,OC,CE)
Collinear(ADB)
Collinear(OEB)
Collinear(DEC)
Conditions (CDL):
CongruentBetweenTriangle(AOB,DOC)
Equal(MeasureOfAngle(AOD),30)
Goal: Value(MeasureOfAngle(ODA))
Answer: 75
Theorem sequence:
congruent_triangle_property_line_equal(1,BAO,CDO)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,AOD)
Problem: As shown in the diagram, ∠ECF=80°. Find the sum of the measure of ∠BEF and the measure of ∠EFA.
问题：如图所示，∠ECF=80°。求∠BEF的大小与∠EFA的大小之和。
Construction (CDL):
Shape(BE,EF,FA,AB)
Shape(EC,CF,FE)
Collinear(BEC)
Collinear(CFA)
Conditions (CDL):
Equal(MeasureOfAngle(ECF),80)
Goal: Value(Add(MeasureOfAngle(BEF),MeasureOfAngle(EFA)))
Answer: 260
Theorem sequence:
triangle_property_angle_sum(1,CFE)
flat_angle(1,BEC)
flat_angle(1,CFA)
angle_addition(1,BEF,FEC)
angle_addition(1,CFE,EFA)
Problem: As shown in the diagram, Add(MeasureOfAngle(EMP)=MeasureOfAngle(QNF)), ∠MPG=108°. Find the measure of ∠PQD.
问题：如图所示，Add(MeasureOfAngle(EMP)=MeasureOfAngle(QNF))，∠MPG=108°。求∠PQD的大小。
Construction (CDL):
Shape(MN,NQ,QP,PM)
Shape(AM,ME)
Shape(EM,MP)
Shape(MP,PG)
Shape(GP,PB)
Shape(NM,MA)
Shape(PM,MN)
Shape(QP,PM)
Shape(BP,PQ)
Shape(CN,NM)
Shape(MN,NQ)
Shape(NQ,QP)
Shape(PQ,QD)
Shape(FN,NC)
Shape(QN,NF)
Shape(HQ,QN)
Shape(DQ,QH)
Collinear(AMPB)
Collinear(CNQD)
Collinear(EMNF)
Collinear(GPQH)
Conditions (CDL):
Equal(Add(MeasureOfAngle(EMP),MeasureOfAngle(QNF)),180)
Equal(MeasureOfAngle(MPG),108)
Goal: Value(MeasureOfAngle(PQD))
Answer: 72
Theorem sequence:
flat_angle(1,EMN)
angle_addition(1,EMP,PMN)
parallel_judgment_corresponding_angle(2,MP,NQ,F)
parallel_property_collinear_extend(1,QN,PM,D)
parallel_property_collinear_extend(2,MP,QD,B)
parallel_property_ipsilateral_internal_angle(1,PB,QD)
vertical_angle(1,MPG,BPQ)
Problem: As shown in the diagram, ∠DEA=34°, AB is parallel to DC, CE is perpendicular to DE. Find the measure of ∠DCE.
问题：如图所示，∠DEA=34°，AB∥DC，CE垂直于DE。求∠DCE的大小。
Construction (CDL):
Shape(ED,DC,CE)
Shape(DE,EA)
Shape(BE,EC)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(DEA),34)
ParallelBetweenLine(AB,DC)
PerpendicularBetweenLine(CE,DE)
Goal: Value(MeasureOfAngle(DCE))
Answer: 56
Theorem sequence:
parallel_property_collinear_extend(3,AB,DC,E)
parallel_property_alternate_interior_angle(2,AE,DC)
triangle_property_angle_sum(1,EDC)
Problem: As shown in the diagram, ∠ABC=30°, AD bisects ∠EAC, AD∥BC. Find the measure of ∠BCA.
问题：如图所示，∠ABC=30°，AD是∠EAC的角平分线，AD平行于BC。求∠BCA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(EA,AD)
Shape(DA,AC)
Collinear(BAE)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),30)
IsBisectorOfAngle(AD,EAC)
ParallelBetweenLine(AD,BC)
Goal: Value(MeasureOfAngle(BCA))
Answer: 30
Theorem sequence:
parallel_property_corresponding_angle(1,AD,BC,E)
parallel_property_alternate_interior_angle(1,AD,BC)
Problem: As shown in the diagram, ∠ACE=110°, ∠CAB=30°, BA is parallel to ED, DE is parallel to FC, FC is parallel to AB. Find the measure of ∠CED.
问题：如图所示，∠ACE=110°，∠CAB=30°，BA平行于ED，DE∥FC，FC平行于AB。求∠CED的大小。
Construction (CDL):
Shape(CE,ED)
Shape(FC,CE)
Shape(AC,CF)
Shape(CA,AB)
Shape(AC,CE)
Conditions (CDL):
Equal(MeasureOfAngle(ACE),110)
Equal(MeasureOfAngle(CAB),30)
ParallelBetweenLine(BA,ED)
ParallelBetweenLine(DE,FC)
ParallelBetweenLine(FC,AB)
Goal: Value(MeasureOfAngle(CED))
Answer: 100
Theorem sequence:
parallel_property_alternate_interior_angle(2,FC,AB)
angle_addition(1,ACF,FCE)
parallel_property_ipsilateral_internal_angle(1,CF,ED)
Problem: As shown in the diagram, ∠HFO=60°, ∠OGE=40°, AB is parallel to CD. Find the measure of ∠HOE.
问题：如图所示，∠HFO=60°，∠OGE=40°，AB平行于CD。求∠HOE的大小。
Construction (CDL):
Shape(GE,EO,OG)
Shape(OH,HF,FO)
Shape(HO,OE)
Shape(GO,OF)
Collinear(AEGB)
Collinear(CHFD)
Collinear(EOF)
Collinear(GOH)
Conditions (CDL):
Equal(MeasureOfAngle(HFO),60)
Equal(MeasureOfAngle(OGE),40)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(HOE))
Answer: 100
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,G)
parallel_property_collinear_extend(3,DC,GA,H)
parallel_property_alternate_interior_angle(2,DH,GA)
triangle_property_angle_sum(1,OHF)
flat_angle(1,FOE)
angle_addition(1,FOH,HOE)
Problem: As shown in the diagram, ∠BCD=40°, O is the center of ⊙O, the tangent to circle O is CB. Find the measure of ∠AEF.
问题：如图所示，∠BCD=40°，⊙O的圆心为O，CB是圆O的切线。求∠AEF的大小。
Construction (CDL):
Shape(OAE,EA)
Shape(OEB,BO,OF,FE)
Shape(OBD,DO,OB)
Shape(ODA,AD)
Shape(AE,EF,FA)
Shape(AF,FD,DA)
Shape(OD,DF,FO)
Shape(OBD,BC,CD)
Collinear(EFD)
Collinear(AFOB)
Collinear(ADC)
Cocircular(O,AEBD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),40)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CB,O)
Goal: Value(MeasureOfAngle(AEF))
Answer: 40
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CB,O,O)
triangle_property_angle_sum(1,ABC)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,AOD)
arc_property_center_angle(1,ODA,O)
arc_property_circumference_angle_external(1,ODA,E)
Problem: As shown in the diagram, AB=AC, ∠EAD=40°, ED perpendicular bisects AB. Find the measure of ∠EBC.
问题：如图所示，AB=AC，∠EAD=40°，ED垂直平分AB。求∠EBC的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DB,BE)
Shape(CE,EB,BC)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(EAD),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)
mirror_congruent_triangle_judgment_sas(1,DEA,DBE)
mirror_congruent_triangle_property_angle_equal(1,ADE,BED)
angle_addition(1,DBE,EBC)
Problem: As shown in the diagram, ∠DAB=36°, BD is the altitude of triangle BCA, AB and AC are the legs of the isosceles △ ABC. Find the measure of ∠DBC.
问题：如图所示，∠DAB=36°，BD是三角形BCA的高，△ABC为等腰△。求∠DBC的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Collinear(ADC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),36)
IsAltitudeOfTriangle(BD,BCA)
IsoscelesTriangle(ABC)
Goal: Value(MeasureOfAngle(DBC))
Answer: 18
Theorem sequence:
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,BCD)
Problem: As shown in the diagram, ∠BCD=120°, O is the center of ⊙O, PD is the tangent to ⊙O. Find the measure of ∠ADP.
问题：如图所示，∠BCD=120°，⊙O的圆心为O，圆O的切线为PD。求∠ADP的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(BC,CD,DO,OB)
Shape(OD,DA,AO)
Shape(OAB,BO,OA)
Shape(ODA,DP,PA)
Collinear(PAOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),120)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PD,O)
Goal: Value(MeasureOfAngle(ADP))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_internal(1,OBD,C)
arc_property_center_angle(1,OBD,O)
flat_angle(1,AOB)
angle_addition(1,AOD,DOB)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,ODA)
tangent_of_circle_property_perpendicular(1,PD,O,O)
angle_addition(1,ODA,ADP)
Problem: As shown in the diagram, ∠ODE=50°, the center of ⊙O is O, the diameter of circle O is CD, AO is parallel to DE. Find the measure of ∠OCA.
问题：如图所示，∠ODE=50°，⊙O的圆心为O，圆O的直径为CD，AO平行于DE。求∠OCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAD,DO,OA)
Shape(ODE,ED)
Shape(OC,CA,AO)
Shape(OEC,CO,OD,DE)
Collinear(COD)
Cocircular(O,ADEC)
Conditions (CDL):
Equal(MeasureOfAngle(ODE),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
ParallelBetweenLine(AO,DE)
Goal: Value(MeasureOfAngle(OCA))
Answer: 25
Theorem sequence:
parallel_property_alternate_interior_angle(2,AO,DE)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,C)
Problem: As shown in the diagram, AB=8, AD=3, △CFA is mirror congruent to △CAB, quadrilateral ABCD is a rectangle. Find the sum of the perimeter of △DAE and the perimeter of triangle FEC.
问题：如图所示，AB=8，AD=3，三角形CFA与三角形CAB是镜像全等三角形，ABCD是矩形。求三角形DAE的周长与三角形FEC的周长之和。
Construction (CDL):
Shape(FE,EC,CF)
Shape(ED,DA,AE)
Shape(CE,EA,AC)
Shape(AB,BC,CA)
Collinear(DEC)
Collinear(AEF)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(AD),3)
MirrorCongruentBetweenTriangle(CFA,CAB)
Rectangle(ABCD)
Goal: Value(Add(PerimeterOfTriangle(DAE),PerimeterOfTriangle(FEC)))
Answer: 22
Theorem sequence:
mirror_congruent_triangle_property_line_equal(1,CFA,CAB)
mirror_congruent_triangle_property_line_equal(1,ACF,ABC)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,DE,EC)
line_addition(1,AE,EF)
triangle_perimeter_formula(1,FEC)
triangle_perimeter_formula(1,EDA)
Problem: As shown in the diagram, OA=OB, OC=OD, ∠AOB=60°, ∠ECA=25°. Find the measure of ∠DEB.
问题：如图所示，OA=OB，OC=OD，∠AOB=60°，∠ECA=25°。求∠DEB的大小。
Construction (CDL):
Shape(EC,CA,AE)
Shape(DE,EB,BD)
Shape(EA,AO,OB,BE)
Collinear(CEB)
Collinear(AED)
Collinear(CAO)
Collinear(OBD)
Conditions (CDL):
Equal(LengthOfLine(OA),LengthOfLine(OB))
Equal(LengthOfLine(OC),LengthOfLine(OD))
Equal(MeasureOfAngle(AOB),60)
Equal(MeasureOfAngle(ECA),25)
Goal: Value(MeasureOfAngle(DEB))
Answer: 70
Theorem sequence:
triangle_property_angle_sum(1,COB)
mirror_congruent_triangle_judgment_sas(1,OBC,ODA)
mirror_congruent_triangle_property_angle_equal(1,COB,DAO)
flat_angle(1,OBD)
angle_addition(1,OBE,EBD)
triangle_property_angle_sum(1,DEB)
Problem: As shown in the diagram, ∠GHE=130°, EFGH is mirror congruent to EHCD, ABCD is a rectangle. Find the measure of ∠BIE.
问题：如图所示，∠GHE=130°，四边形EFGH与四边形EHCD镜像全等，四边形ABCD是矩形。求∠BIE的大小。
Construction (CDL):
Shape(AB,BI,IE,EA)
Shape(DE,EH,HC,CD)
Shape(EI,IH,HE)
Shape(IF,FG,GH,HI)
Collinear(AED)
Collinear(BIHC)
Collinear(FIE)
Conditions (CDL):
Equal(MeasureOfAngle(GHE),130)
MirrorCongruentBetweenQuadrilateral(EFGH,EHCD)
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(BIE))
Answer: 100
Theorem sequence:
parallel_judgment_ipsilateral_internal_angle(1,AE,BI)
parallel_judgment_ipsilateral_internal_angle(1,CH,DE)
parallel_property_ipsilateral_internal_angle(1,IB,EA)
parallel_property_ipsilateral_internal_angle(1,ED,HC)
mirror_congruent_quadrilateral_property_angle_equal(1,EFGH,EHCD)
mirror_congruent_quadrilateral_property_angle_equal(1,HEFG,HCDE)
angle_addition(1,DEH,HEI)
flat_angle(1,DEA)
angle_addition(1,DEI,IEA)
Problem: As shown in the diagram, ∠CPM=43°, ∠NPD=∠GNB. Find the measure of ∠EMN.
问题：如图所示，∠CPM=43°，∠NPD=∠GNB。求∠EMN的大小。
Construction (CDL):
Shape(NM,MP,PN)
Shape(AM,ME)
Shape(EM,MN)
Shape(MN,NG)
Shape(GN,NB)
Shape(PM,MA)
Shape(BN,NP)
Shape(CP,PM)
Shape(NP,PD)
Shape(HP,PC)
Shape(FP,PH)
Shape(DP,PF)
Shape(MP,PD)
Collinear(AMNB)
Collinear(CPD)
Collinear(EMPF)
Collinear(GNPH)
Conditions (CDL):
Equal(MeasureOfAngle(CPM),43)
Equal(MeasureOfAngle(NPD),MeasureOfAngle(GNB))
Goal: Value(MeasureOfAngle(EMN))
Answer: 137
Theorem sequence:
parallel_judgment_corresponding_angle(1,NB,PD,G)
parallel_property_collinear_extend(1,NB,PD,M)
parallel_property_corresponding_angle(1,MN,PD,E)
flat_angle(1,CPD)
angle_addition(1,CPM,MPD)
Problem: As shown in the diagram, ∠AGE=128°, HM bisects ∠GHD, AB is parallel to CD. Find the measure of ∠MHD.
问题：如图所示，∠AGE=128°，HM是∠GHD的角平分线，AB平行于CD。求∠MHD的大小。
Construction (CDL):
Shape(AG,GE)
Shape(EG,GB)
Shape(HG,GA)
Shape(BG,GH)
Shape(CH,HG)
Shape(GH,HM)
Shape(MH,HD)
Shape(GH,HD)
Shape(FH,HC)
Shape(DH,HF)
Collinear(AGB)
Collinear(CHD)
Collinear(EGHF)
Conditions (CDL):
Equal(MeasureOfAngle(AGE),128)
IsBisectorOfAngle(HM,GHD)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(MHD))
Answer: 26
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,G)
parallel_property_collinear_extend(3,DC,GA,H)
parallel_property_corresponding_angle(2,HC,GA,E)
flat_angle(1,CHD)
angle_addition(1,CHG,GHD)
angle_addition(1,GHM,MHD)
Problem: As shown in the diagram, ∠DAB=115°, AD∥BC. Find the measure of ∠ABC.
问题：如图所示，∠DAB=115°，AD平行于BC。求∠ABC的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),115)
ParallelBetweenLine(AD,BC)
Goal: Value(MeasureOfAngle(ABC))
Answer: 65
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
Problem: As shown in the diagram, ∠ACD=150°, ∠CAB=80°. Find the measure of ∠ABC.
问题：如图所示，∠ACD=150°，∠CAB=80°。求∠ABC的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD)
Collinear(BCD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),150)
Equal(MeasureOfAngle(CAB),80)
Goal: Value(MeasureOfAngle(ABC))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,BCA,ACD)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, AB=8, BC=10, △AFE is mirror congruent to △AED, ABCD is a rectangle. Find the perimeter of triangle CEF.
问题：如图所示，AB=8，BC=10，三角形AFE与三角形AED是镜像全等三角形，ABCD是矩形。求△CEF的周长。
Construction (CDL):
Shape(AB,BF,FA)
Shape(AF,FE,EA)
Shape(AE,ED,DA)
Shape(FC,CE,EF)
Collinear(BFC)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(BC),10)
MirrorCongruentBetweenTriangle(AFE,AED)
Rectangle(ABCD)
Goal: Value(PerimeterOfTriangle(CEF))
Answer: 12
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
mirror_congruent_triangle_property_line_equal(1,AFE,AED)
mirror_congruent_triangle_property_line_equal(1,EAF,EDA)
right_triangle_judgment_angle(1,ABF)
right_triangle_property_pythagorean(1,ABF)
line_addition(1,BF,FC)
line_addition(1,DE,EC)
triangle_perimeter_formula(1,FCE)
Problem: As shown in the diagram, ∠COD=20°, triangle AOD is a right triangle, ∠COB is the right angle of the right triangle COB. Find the measure of ∠AOB.
问题：如图所示，∠COD=20°，AO和OD是直角三角形AOD的两个直角边，∠COB是直角三角形COB的直角。求∠AOB的大小。
Construction (CDL):
Shape(AO,OE,EA)
Shape(CE,EF,FC)
Shape(FE,EO,OG,GF)
Shape(DF,FG,GD)
Shape(GO,OB,BG)
Collinear(AEFD)
Collinear(OGD)
Collinear(OEC)
Collinear(OGD)
Collinear(CFGB)
Conditions (CDL):
Equal(MeasureOfAngle(COD),20)
RightTriangle(AOD)
RightTriangle(COB)
Goal: Value(MeasureOfAngle(AOB))
Answer: 160
Theorem sequence:
angle_addition(1,AOE,EOG)
angle_addition(1,AOE,EOB)
Problem: As shown in the diagram, ∠BAC=60°, ∠CBA=30°, ∠CDE=40°, DE∥FA, △ACB is a right △. Find the measure of ∠BAF.
问题：如图所示，∠BAC=60°，∠CBA=30°，∠CDE=40°，DE∥FA，AC和CB是直角三角形ACB的两个直角边。求∠BAF的大小。
Construction (CDL):
Shape(CD,DE,EC)
Shape(DF,FA,AE,ED)
Shape(FB,BA,AF)
Collinear(CDFB)
Collinear(CEA)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),60)
Equal(MeasureOfAngle(CBA),30)
Equal(MeasureOfAngle(CDE),40)
ParallelBetweenLine(DE,FA)
RightTriangle(ACB)
Goal: Value(MeasureOfAngle(BAF))
Answer: 10
Theorem sequence:
triangle_property_angle_sum(1,CDE)
parallel_property_corresponding_angle(2,AF,ED,C)
angle_addition(1,BAF,FAC)
Problem: As shown in the diagram, AB=AC, ∠CAB=30°, DI perpendicular bisects AB. Find the measure of ∠DBC.
问题：如图所示，AB=AC，∠CAB=30°，DI垂直平分AB。求∠DBC的大小。
Construction (CDL):
Shape(AI,ID,DA)
Shape(IB,BD,DI)
Shape(DB,BC,CD)
Collinear(AIB)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),30)
IsPerpendicularBisectorOfLine(DI,AB)
Goal: Value(MeasureOfAngle(DBC))
Answer: 45
Theorem sequence:
mirror_congruent_triangle_judgment_sas(1,IDA,IBD)
mirror_congruent_triangle_property_angle_equal(1,AID,BDI)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
angle_addition(1,ABD,DBC)
Problem: As shown in the diagram, ∠ADE=30°, ∠ABC and ∠BCA are the base angles of the isosceles right triangle ABC, DE∥FG. Find the measure of ∠GFC.
问题：如图所示，∠ADE=30°，∠CAB是等腰直角三角形ABC的直角，DE∥FG。求∠GFC的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BF,FG,GE,ED)
Shape(GF,FC,CG)
Collinear(ADB)
Collinear(AEGC)
Collinear(BFC)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),30)
IsoscelesRightTriangle(ABC)
ParallelBetweenLine(DE,FG)
Goal: Value(MeasureOfAngle(GFC))
Answer: 15
Theorem sequence:
triangle_property_angle_sum(1,ADE)
parallel_property_corresponding_angle(2,GF,ED,A)
adjacent_complementary_angle(1,CGF,FGE)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,GFC)
Problem: As shown in the diagram, ∠ACE=70°, AE is the angle bisector of ∠BAC, AB is parallel to CD. Find the measure of ∠AED.
问题：如图所示，∠ACE=70°，AE是∠BAC的角平分线，AB∥CD。求∠AED的大小。
Construction (CDL):
Shape(AC,CE,EA)
Shape(AE,ED,DB,BA)
Collinear(CED)
Conditions (CDL):
Equal(MeasureOfAngle(ACE),70)
IsBisectorOfAngle(AE,BAC)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(AED))
Answer: 125
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AB,CD)
angle_addition(1,BAE,EAC)
triangle_property_angle_sum(1,ACE)
adjacent_complementary_angle(1,CEA,AED)
Problem: As shown in the diagram, ∠ACD=30°, the center of ⊙O is O, BA is the diameter of ⊙O. Find the measure of ∠DAB.
问题：如图所示，∠ACD=30°，圆O的圆心为O，BA是圆O的直径。求∠DAB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AE,ED)
Shape(OAC,CA)
Shape(EA,AC,CE)
Shape(EC,OCB,BO,OE)
Shape(DE,EO,OB,BD)
Shape(OBD,DB)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BA,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 60
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠ACD=35°, ∠BCA=90°, BA∥CD. Find the measure of ∠ABC.
问题：如图所示，∠ACD=35°，∠BCA=90°，BA∥CD。求∠ABC的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),35)
Equal(MeasureOfAngle(BCA),90)
ParallelBetweenLine(BA,CD)
Goal: Value(MeasureOfAngle(ABC))
Answer: 55
Theorem sequence:
angle_addition(1,BCA,ACD)
parallel_property_ipsilateral_internal_angle(1,BA,CD)
Problem: As shown in the diagram, AC=8, BD=6, quadrilateral ABCD is a rhombus. Find the perimeter of △ABC.
问题：如图所示，AC=8，BD=6，ABCD是菱形。求三角形ABC的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(OB,BC,CO)
Shape(DO,OC,CD)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(BD),6)
Rhombus(ABCD)
Goal: Value(PerimeterOfTriangle(ABC))
Answer: 18
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
kite_property_diagonal_perpendicular_bisection(1,BCDA,O)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
right_triangle_judgment_angle(1,BOA)
right_triangle_property_pythagorean(1,BOA)
triangle_perimeter_formula(1,ABC)
Problem: As shown in the diagram, ∠GCF=25°, FC bisects ∠EFA, AB∥CD, CD∥EF. Find the measure of ∠BAF.
问题：如图所示，∠GCF=25°，FC是∠EFA的角平分线，AB∥CD，CD平行于EF。求∠BAF的大小。
Construction (CDL):
Shape(AC,CG,GA)
Shape(AG,GD,DB,BA)
Shape(GC,CF,FG)
Shape(GF,FD,DG)
Shape(CE,EF,FC)
Collinear(AGF)
Collinear(CGD)
Collinear(BDF)
Conditions (CDL):
Equal(MeasureOfAngle(GCF),25)
IsBisectorOfAngle(FC,EFA)
ParallelBetweenLine(AB,CD)
ParallelBetweenLine(CD,EF)
Goal: Value(MeasureOfAngle(BAF))
Answer: 50
Theorem sequence:
parallel_property_alternate_interior_angle(1,CD,EF)
parallel_judgment_par_par(1,AB,CD,EF)
parallel_property_alternate_interior_angle(1,AB,EF)
angle_addition(1,EFC,CFA)
Problem: As shown in the diagram, ∠AGE=115°, ∠CAB=30°, DE∥FC, △BCA is a right △. Find the measure of ∠DFC.
问题：如图所示，∠AGE=115°，∠CAB=30°，DE∥FC，△BCA为直角△。求∠DFC的大小。
Construction (CDL):
Shape(AD,DG,GA)
Shape(DF,FC,CG,GD)
Shape(FB,BC,CF)
Shape(AG,GE)
Shape(EG,GC)
Collinear(ADFB)
Collinear(AGC)
Collinear(DGE)
Conditions (CDL):
Equal(MeasureOfAngle(AGE),115)
Equal(MeasureOfAngle(CAB),30)
ParallelBetweenLine(DE,FC)
RightTriangle(BCA)
Goal: Value(MeasureOfAngle(DFC))
Answer: 85
Theorem sequence:
adjacent_complementary_angle(1,DGA,AGE)
triangle_property_angle_sum(1,ADG)
parallel_property_corresponding_angle(1,DE,FC,A)
Problem: As shown in the diagram, ∠GEB=60°, AB∥CD. Find the measure of ∠HFC.
问题：如图所示，∠GEB=60°，AB∥CD。求∠HFC的大小。
Construction (CDL):
Shape(AE,EG)
Shape(GE,EB)
Shape(BE,EF)
Shape(FE,EA)
Shape(CF,FE)
Shape(EF,FD)
Shape(DF,FH)
Shape(HF,FC)
Collinear(AEB)
Collinear(CFD)
Collinear(GEFH)
Conditions (CDL):
Equal(MeasureOfAngle(GEB),60)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(HFC))
Answer: 60
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,E)
parallel_property_collinear_extend(3,DC,BE,F)
adjacent_complementary_angle(1,GEB,BEF)
parallel_property_ipsilateral_internal_angle(1,EB,FD)
vertical_angle(1,EFD,HFC)
Problem: As shown in the diagram, ∠BCA=30°, ∠CGF=25°, DE∥FG, AB and BC are the legs of the right △ ABC. Find the measure of ∠BDE.
问题：如图所示，∠BCA=30°，∠CGF=25°，DE∥FG，∠ABC是直角三角形ABC的直角。求∠BDE的大小。
Construction (CDL):
Shape(BD,DE,EA,AB)
Shape(DF,FG,GE,ED)
Shape(FC,CG,GF)
Collinear(BDFC)
Collinear(AEGC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),30)
Equal(MeasureOfAngle(CGF),25)
ParallelBetweenLine(DE,FG)
RightTriangle(ABC)
Goal: Value(MeasureOfAngle(BDE))
Answer: 55
Theorem sequence:
triangle_property_angle_sum(1,FCG)
adjacent_complementary_angle(1,DFG,GFC)
parallel_property_corresponding_angle(1,DE,FG,B)
Problem: As shown in the diagram, ∠CAB=66°, I is the center of the inscribed circle of △ABC. Find the measure of ∠CIB.
问题：如图所示，∠CAB=66°，I是三角形ABC内切圆的圆心。求∠CIB的大小。
Construction (CDL):
Shape(AB,BI,IC,CA)
Shape(IB,BC,CI)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),66)
IsIncenterOfTriangle(I,ABC)
Goal: Value(MeasureOfAngle(CIB))
Answer: 123
Theorem sequence:
triangle_property_angle_sum(1,ABC)
angle_addition(1,ABI,IBC)
angle_addition(1,BCI,ICA)
triangle_property_angle_sum(1,IBC)
Problem: As shown in the diagram, ∠EBC=35°, CD is parallel to EF, CB⊥AB. Find the measure of ∠AGD.
问题：如图所示，∠EBC=35°，CD∥EF，CB垂直于AB。求∠AGD的大小。
Construction (CDL):
Shape(CE,EB,BC)
Shape(CB,BG,GC)
Shape(AG,GD,DA)
Shape(GB,BF,FD,DG)
Collinear(CGD)
Collinear(EBF)
Collinear(AGB)
Collinear(ADF)
Conditions (CDL):
Equal(MeasureOfAngle(EBC),35)
ParallelBetweenLine(CD,EF)
PerpendicularBetweenLine(CB,AB)
Goal: Value(MeasureOfAngle(AGD))
Answer: 55
Theorem sequence:
parallel_property_collinear_extend(3,CD,EF,G)
parallel_property_collinear_extend(3,FE,GC,B)
angle_addition(1,EBC,CBG)
parallel_property_ipsilateral_internal_angle(1,BE,GC)
vertical_angle(1,AGD,BGC)
Problem: As shown in the diagram, ∠IAF=30°, AC and AB are the legs of the isosceles right triangle ACB, DEFG is a rectangle. Find the measure of ∠DHB.
问题：如图所示，∠IAF=30°，∠BAC是等腰直角△ACB的直角，四边形DEFG是矩形。求∠DHB的大小。
Construction (CDL):
Shape(BH,HI,IC,CB)
Shape(HA,AI,IH)
Shape(DE,EA,AH,HD)
Shape(IA,AF,FG,GI)
Shape(DH,HB)
Shape(CI,IG)
Collinear(DHIG)
Collinear(EAF)
Collinear(BHA)
Collinear(CIA)
Conditions (CDL):
Equal(MeasureOfAngle(IAF),30)
IsoscelesRightTriangle(ACB)
Rectangle(DEFG)
Goal: Value(MeasureOfAngle(DHB))
Answer: 60
Theorem sequence:
angle_addition(1,BAC,CAF)
parallel_property_collinear_extend(3,DG,EF,H)
parallel_property_collinear_extend(3,FE,GH,A)
parallel_property_ipsilateral_internal_angle(1,HG,AF)
vertical_angle(1,DHB,IHA)