SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠GEB=70°, AB is parallel to CD. Find the measure of ∠CFE.
问题：如图所示，∠GEB=70°，AB∥CD。求∠CFE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),70)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(CFE))
Answer: 110
Theorem sequence:
vertical_angle(1,GEB,FEA)
parallel_property_collinear_extend(3,AB,CD,E)
parallel_property_collinear_extend(3,DC,EA,F)
parallel_property_ipsilateral_internal_angle(1,FC,EA)
Problem: As shown in the diagram, ∠ABC=45°, ∠BCA=60°, DF∥AB, FE is parallel to CA. Find the measure of ∠EFD.
问题：如图所示，∠ABC=45°，∠BCA=60°，DF∥AB，FE平行于CA。求∠EFD的大小。
Construction (CDL):
Shape(AE,EF,FD,DA)
Shape(EB,BF,FE)
Shape(DF,FC,CD)
Collinear(AEB)
Collinear(BFC)
Collinear(CDA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),45)
Equal(MeasureOfAngle(BCA),60)
ParallelBetweenLine(DF,AB)
ParallelBetweenLine(FE,CA)
Goal: Value(MeasureOfAngle(EFD))
Answer: 75
Theorem sequence:
parallel_property_corresponding_angle(2,BA,FD,C)
parallel_property_ipsilateral_internal_angle(1,FE,CA)
angle_addition(1,EFD,DFC)
Problem: As shown in the diagram, ∠EAD=35°, AD is parallel to BC, CA⊥BA. Find the measure of ∠BCA.
问题：如图所示，∠EAD=35°，AD∥BC，CA⊥BA。求∠BCA的大小。
Construction (CDL):
Shape(EA,AD,DE)
Shape(AC,CD,DA)
Shape(AB,BC,CA)
Collinear(EAB)
Collinear(EDC)
Conditions (CDL):
Equal(MeasureOfAngle(EAD),35)
ParallelBetweenLine(AD,BC)
PerpendicularBetweenLine(CA,BA)
Goal: Value(MeasureOfAngle(BCA))
Answer: 55
Theorem sequence:
adjacent_complementary_angle(1,EAD,DAB)
angle_addition(1,DAC,CAB)
parallel_property_alternate_interior_angle(1,AD,BC)
Problem: As shown in the diagram, ∠COE=50°, EO⊥BO. Find the measure of ∠BOD.
问题：如图所示，∠COE=50°，EO垂直于BO。求∠BOD的大小。
Construction (CDL):
Shape(CO,OE,EC)
Shape(EO,OB,BE)
Shape(BO,OD,DB)
Collinear(COD)
Conditions (CDL):
Equal(MeasureOfAngle(COE),50)
PerpendicularBetweenLine(EO,BO)
Goal: Value(MeasureOfAngle(BOD))
Answer: 40
Theorem sequence:
angle_addition(1,COE,EOB)
adjacent_complementary_angle(1,COB,BOD)
Problem: As shown in the diagram, ∠DEB=10°, EF bisects ∠BEC, AB is parallel to MF. Find the measure of ∠MFE.
问题：如图所示，∠DEB=10°，EF是∠BEC的角平分线，AB∥MF。求∠MFE的大小。
Construction (CDL):
Shape(AC,CE,EA)
Shape(DE,EB,BD)
Shape(EC,CM,MF,FE)
Shape(EF,FB,BE)
Collinear(ACM)
Collinear(AEB)
Collinear(FBD)
Collinear(CED)
Conditions (CDL):
Equal(MeasureOfAngle(DEB),10)
IsBisectorOfAngle(EF,BEC)
ParallelBetweenLine(AB,MF)
Goal: Value(MeasureOfAngle(MFE))
Answer: 85
Theorem sequence:
vertical_angle(1,DEB,CEA)
angle_addition(1,BEF,FEC)
adjacent_complementary_angle(1,BEC,CEA)
parallel_property_collinear_extend(3,AB,MF,E)
parallel_property_alternate_interior_angle(1,EB,MF)
Problem: As shown in the diagram, △CAB is congruent to △EBD, ∠ABC=100°, ∠CAB=45°. Find the measure of ∠CBE.
问题：如图所示，三角形CAB全等于三角形EBD，∠ABC=100°，∠CAB=45°。求∠CBE的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BD,DE,EB)
Shape(CB,BE)
Collinear(ABD)
Conditions (CDL):
CongruentBetweenTriangle(CAB,EBD)
Equal(MeasureOfAngle(ABC),100)
Equal(MeasureOfAngle(CAB),45)
Goal: Value(MeasureOfAngle(CBE))
Answer: 35
Theorem sequence:
congruent_triangle_property_angle_equal(1,ABC,BDE)
angle_addition(1,ABC,CBE)
adjacent_complementary_angle(1,ABE,EBD)
Problem: As shown in the diagram, ∠CBA=72°, AB is parallel to CD, CB∥DE. Find the measure of ∠CDE.
问题：如图所示，∠CBA=72°，AB∥CD，CB平行于DE。求∠CDE的大小。
Construction (CDL):
Shape(CB,BA)
Shape(BC,CD)
Shape(CD,DE)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),72)
ParallelBetweenLine(AB,CD)
ParallelBetweenLine(CB,DE)
Goal: Value(MeasureOfAngle(CDE))
Answer: 108
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CD)
parallel_property_ipsilateral_internal_angle(1,CB,DE)
Problem: As shown in the diagram, ∠AOD=54°, the center of ⊙O is O, BA is the diameter of circle O. Find the measure of ∠DCB.
问题：如图所示，∠AOD=54°，O是⊙O的圆心，圆O的直径为BA。求∠DCB的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(ODA,AO,OD)
Shape(OA,AC,CO)
Shape(OC,CB,BO)
Shape(OCB,BC)
Shape(OB,OBD,DO)
Collinear(AOB)
Collinear(COD)
Cocircular(O,CBDA)
Conditions (CDL):
Equal(MeasureOfAngle(AOD),54)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BA,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 63
Theorem sequence:
arc_property_center_angle(1,ODA,O)
arc_property_circumference_angle_external(1,ODA,C)
diameter_of_circle_property_right_angle(1,ACB,O)
angle_addition(1,ACD,DCB)
Problem: As shown in the diagram, ∠FNC=120°, AB∥CD. Find the measure of ∠BMN.
问题：如图所示，∠FNC=120°，AB平行于CD。求∠BMN的大小。
Construction (CDL):
Shape(AM,ME)
Shape(EM,MB)
Shape(BM,MN)
Shape(NM,MA)
Shape(MN,ND)
Shape(DN,NF)
Shape(FN,NC)
Shape(CN,NM)
Collinear(AMB)
Collinear(CND)
Collinear(EMNF)
Conditions (CDL):
Equal(MeasureOfAngle(FNC),120)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BMN))
Answer: 60
Theorem sequence:
adjacent_complementary_angle(1,FNC,CNM)
parallel_property_collinear_extend(3,AB,CD,M)
parallel_property_collinear_extend(3,DC,BM,N)
parallel_property_alternate_interior_angle(1,MB,CN)
Problem: As shown in the diagram, AB=5, AC=6, CA∥ED, ABCD is a rhombus. Find the area of △BED.
问题：如图所示，AB=5，AC=6，CA平行于ED，四边形ABCD是菱形。求△BED的面积。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(AO,OD,DA)
Shape(DO,OC,CD)
Shape(DC,CE,ED)
Collinear(AOC)
Collinear(BOD)
Collinear(BCE)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(AC),6)
ParallelBetweenLine(CA,ED)
Rhombus(ABCD)
Goal: Value(AreaOfTriangle(BED))
Answer: 24
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)
parallel_property_collinear_extend(1,CB,DA,E)
parallelogram_judgment_parallel_and_parallel(1,ACED)
parallelogram_property_opposite_line_equal(1,ACED)
parallelogram_property_opposite_line_equal(1,CEDA)
line_addition(1,BC,CE)
right_triangle_judgment_pythagorean_inverse(1,EDB)
triangle_area_formula_sine(1,DBE)
Problem: As shown in the diagram, ∠CEG=80°, ∠GEF=40°, AB is parallel to CD. Find the measure of ∠EFG.
问题：如图所示，∠CEG=80°，∠GEF=40°，AB平行于CD。求∠EFG的大小。
Construction (CDL):
Shape(AC,CE,EG,GA)
Shape(GE,EF,FG)
Shape(FE,ED,DB,BF)
Collinear(AGFB)
Collinear(CED)
Conditions (CDL):
Equal(MeasureOfAngle(CEG),80)
Equal(MeasureOfAngle(GEF),40)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(EFG))
Answer: 60
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,G)
parallel_property_collinear_extend(3,DC,BG,E)
parallel_property_alternate_interior_angle(1,GB,CE)
triangle_property_angle_sum(1,GEF)
Problem: As shown in the diagram, ∠APC=28°, CP is perpendicular to DP. Find the measure of ∠DPB.
问题：如图所示，∠APC=28°，CP垂直于DP。求∠DPB的大小。
Construction (CDL):
Shape(AP,PC)
Shape(CP,PD)
Shape(DP,PB)
Collinear(APB)
Conditions (CDL):
Equal(MeasureOfAngle(APC),28)
PerpendicularBetweenLine(CP,DP)
Goal: Value(MeasureOfAngle(DPB))
Answer: 62
Theorem sequence:
adjacent_complementary_angle(1,APC,CPB)
angle_addition(1,CPD,DPB)
Problem: As shown in the diagram, CA=CD, ∠CAB=25°, O is the center of circle O, BA is the diameter of circle O. Find the measure of ∠DCA.
问题：如图所示，CA=CD，∠CAB=25°，O是⊙O的圆心，⊙O的直径为BA。求∠DCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(EC,CA,AO,OE)
Shape(CE,EB,BC)
Shape(OBC,CB)
Shape(AD,DE,EO,OA)
Shape(BE,ED,ODB)
Shape(DA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(CA),LengthOfLine(CD))
Equal(MeasureOfAngle(CAB),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BA,O)
Goal: Value(MeasureOfAngle(DCA))
Answer: 50
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)
isosceles_triangle_judgment_line_equal(1,CAD)
isosceles_triangle_property_angle_equal(1,CAD)
triangle_property_angle_sum(1,CAD)
Problem: As shown in the diagram, ∠DOA=86°, ∠DOB=30°, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠DOA=86°，∠DOB=30°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OAB,BF,FA)
Shape(OBD,DO,OB)
Shape(OF,FB,BO)
Shape(AF,FC,CA)
Shape(OCA,AC)
Shape(CF,FO,OE,OEC)
Collinear(BFC)
Collinear(AFO)
Collinear(DOE)
Cocircular(O,ECABD)
Conditions (CDL):
Equal(MeasureOfAngle(DOA),86)
Equal(MeasureOfAngle(DOB),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 28
Theorem sequence:
angle_addition(1,DOB,BOA)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠CAB=42°, ∠DPA=77°, O is the center of ⊙O. Find the measure of ∠DBA.
问题：如图所示，∠CAB=42°，∠DPA=77°，圆O的圆心为O。求∠DBA的大小。
Construction (CDL):
Shape(O)
Shape(OBC,CP,PB)
Shape(OCA,AC)
Shape(CA,AP,PC)
Shape(PA,OAD,DP)
Shape(BP,PD,DB)
Shape(BD,ODB)
Collinear(CPD)
Collinear(APB)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),42)
Equal(MeasureOfAngle(DPA),77)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 35
Theorem sequence:
adjacent_complementary_angle(1,DPA,APC)
triangle_property_angle_sum(1,CAP)
arc_property_circumference_angle_external(1,OAD,C)
arc_property_circumference_angle_external(1,OAD,B)
Problem: As shown in the diagram, ∠CBA=50°, AB is the diameter of ⊙O, D bisects arc OAC. Find the measure of ∠BAD.
问题：如图所示，∠CBA=50°，⊙O的直径为AB，D平分⌒OAC。求∠BAD的大小。
Construction (CDL):
Shape(OCB,BC)
Shape(ODC,CD)
Shape(OAD,DA)
Shape(CB,BD,DC)
Shape(DB,BA,AD)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),50)
IsDiameterOfCircle(AB,O)
IsMidpointOfArc(D,OAC)
Goal: Value(MeasureOfAngle(BAD))
Answer: 65
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAD,ODC)
congruent_arc_property_measure_equal(1,OAD,ODC)
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,ODC,B)
angle_addition(1,CBD,DBA)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,BAD)
Problem: As shown in the diagram, ∠BCD=140°, AB∥EF, CD is perpendicular to FD, DF is perpendicular to BF. Find the measure of ∠CBA.
问题：如图所示，∠BCD=140°，AB∥EF，CD⊥FD，DF⊥BF。求∠CBA的大小。
Construction (CDL):
Shape(AE,ED,DC,CB,BA)
Shape(BC,CD,DF,FB)
Collinear(EDF)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),140)
ParallelBetweenLine(AB,EF)
PerpendicularBetweenLine(CD,FD)
PerpendicularBetweenLine(DF,BF)
Goal: Value(MeasureOfAngle(CBA))
Answer: 50
Theorem sequence:
quadrilateral_property_angle_sum(1,BCDF)
parallel_property_ipsilateral_internal_angle(1,FE,BA)
angle_addition(1,FBC,CBA)
Problem: As shown in the diagram, ∠MND=60°, AB is parallel to CD. Find the measure of ∠AME.
问题：如图所示，∠MND=60°，AB∥CD。求∠AME的大小。
Construction (CDL):
Shape(AM,ME)
Shape(EM,MB)
Shape(BM,MN)
Shape(NM,MA)
Shape(MN,ND)
Shape(DN,NF)
Shape(FN,NC)
Shape(CN,NM)
Collinear(AMB)
Collinear(CND)
Collinear(FNME)
Conditions (CDL):
Equal(MeasureOfAngle(MND),60)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(AME))
Answer: 120
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,M)
parallel_property_collinear_extend(3,DC,MA,N)
parallel_property_alternate_interior_angle(2,AM,ND)
adjacent_complementary_angle(1,NMA,AME)
Problem: As shown in the diagram, ∠ABC=40°, ∠ACD=65°, DC∥AB. Find the measure of ∠BCA.
问题：如图所示，∠ABC=40°，∠ACD=65°，DC∥AB。求∠BCA的大小。
Construction (CDL):
Shape(DA,AC,CD)
Shape(CA,AB,BC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),40)
Equal(MeasureOfAngle(ACD),65)
ParallelBetweenLine(DC,AB)
Goal: Value(MeasureOfAngle(BCA))
Answer: 75
Theorem sequence:
parallel_property_alternate_interior_angle(2,DC,AB)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠ABC=45°, ∠FDE=30°, ∠CAB is the right angle of the right triangle CAB, ∠DEF is the right angle of the right triangle DEF. Find the measure of ∠DMB.
问题：如图所示，∠ABC=45°，∠FDE=30°，∠CAB是直角△CAB的直角，DE和EF是直角△DEF的两个直角边。求∠DMB的大小。
Construction (CDL):
Shape(NB,BM,MN)
Shape(AF,FN,NA)
Shape(AN,NM,ML,LE,EA)
Shape(MD,DL,LM)
Shape(EL,LC,CE)
Shape(BN,NF)
Shape(DM,MB)
Shape(CL,LD)
Collinear(FAEC)
Collinear(FNMD)
Collinear(ELD)
Collinear(ANB)
Collinear(BMLC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),45)
Equal(MeasureOfAngle(FDE),30)
RightTriangle(CAB)
RightTriangle(DEF)
Goal: Value(MeasureOfAngle(DMB))
Answer: 75
Theorem sequence:
parallel_judgment_ipsilateral_internal_angle(1,ED,AB)
parallel_property_collinear_extend(3,BA,DE,N)
parallel_property_alternate_interior_angle(2,BN,DE)
triangle_property_angle_sum(1,NBM)
adjacent_complementary_angle(1,DMB,BMN)
Problem: As shown in the diagram, ∠CFE=120°, AB∥CD, GE is perpendicular to FE. Find the measure of ∠BEG.
问题：如图所示，∠CFE=120°，AB平行于CD，GE垂直于FE。求∠BEG的大小。
Construction (CDL):
Shape(AC,CF,FE,EA)
Shape(EF,FD,DG,GE)
Shape(EG,GB,BE)
Collinear(AEB)
Collinear(CFD)
Conditions (CDL):
Equal(MeasureOfAngle(CFE),120)
ParallelBetweenLine(AB,CD)
PerpendicularBetweenLine(GE,FE)
Goal: Value(MeasureOfAngle(BEG))
Answer: 30
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,E)
parallel_property_collinear_extend(3,DC,BE,F)
parallel_property_alternate_interior_angle(1,EB,CF)
angle_addition(1,BEG,GEF)
Problem: As shown in the diagram, ∠BME=48°, DEFG is a rectangle, CA and AB are the legs of the right △ CAB. Find the measure of ∠GAN.
问题：如图所示，∠BME=48°，DEFG是长方形，∠CAB是直角△CAB的直角。求∠GAN的大小。
Construction (CDL):
Shape(DE,EM,MA,AD)
Shape(AM,MN,NA)
Shape(AN,NF,FG,GA)
Shape(MB,BC,CN,NM)
Shape(BM,ME)
Shape(FN,NC)
Collinear(DAG)
Collinear(EMNF)
Collinear(AMB)
Collinear(ANC)
Conditions (CDL):
Equal(MeasureOfAngle(BME),48)
Rectangle(DEFG)
RightTriangle(CAB)
Goal: Value(MeasureOfAngle(GAN))
Answer: 42
Theorem sequence:
parallel_property_collinear_extend(3,DG,EF,A)
parallel_property_collinear_extend(3,FE,AD,M)
parallel_property_corresponding_angle(1,ME,AD,B)
adjacent_complementary_angle(1,GAM,MAD)
angle_addition(1,GAN,NAM)
Problem: As shown in the diagram, ∠FBA=50°, ∠FCE=40°, DC∥BA. Find the measure of ∠CEF.
问题：如图所示，∠FBA=50°，∠FCE=40°，DC平行于BA。求∠CEF的大小。
Construction (CDL):
Shape(DF,FE)
Shape(EF,FC,CE)
Shape(BF,FD)
Shape(CF,FB)
Shape(FB,BA)
Collinear(DFC)
Collinear(EFB)
Conditions (CDL):
Equal(MeasureOfAngle(FBA),50)
Equal(MeasureOfAngle(FCE),40)
ParallelBetweenLine(DC,BA)
Goal: Value(MeasureOfAngle(CEF))
Answer: 90
Theorem sequence:
parallel_property_collinear_extend(3,DC,BA,F)
parallel_property_corresponding_angle(1,FC,BA,E)
triangle_property_angle_sum(1,EFC)
Problem: As shown in the diagram, ∠EBA=37°, AB∥CD, BE is perpendicular to DE. Find the measure of ∠CDE.
问题：如图所示，∠EBA=37°，AB∥CD，BE垂直于DE。求∠CDE的大小。
Construction (CDL):
Shape(AC,CD,DE,EB,BA)
Shape(BE,ED,DB)
Conditions (CDL):
Equal(MeasureOfAngle(EBA),37)
ParallelBetweenLine(AB,CD)
PerpendicularBetweenLine(BE,DE)
Goal: Value(MeasureOfAngle(CDE))
Answer: 53
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,DC,BA)
angle_addition(1,DBE,EBA)
angle_addition(1,CDE,EDB)
triangle_property_angle_sum(1,BED)
Problem: As shown in the diagram, ∠CHF=110°, ∠FEG=50°, AB∥CD. Find the measure of ∠EGF.
问题：如图所示，∠CHF=110°，∠FEG=50°，AB平行于CD。求∠EGF的大小。
Construction (CDL):
Shape(AC,CH,HG,GE,EA)
Shape(EG,GF,FE)
Shape(FG,GH,HD,DB,BF)
Collinear(AEFB)
Collinear(CHD)
Collinear(HGF)
Conditions (CDL):
Equal(MeasureOfAngle(CHF),110)
Equal(MeasureOfAngle(FEG),50)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(EGF))
Answer: 60
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,F)
parallel_property_collinear_extend(3,DC,FA,H)
parallel_property_ipsilateral_internal_angle(1,HC,FA)
triangle_property_angle_sum(1,EGF)
Problem: As shown in the diagram, ∠ABC=30°, ∠EAD=55°, AD is the angle bisector of ∠EAC. Find the measure of ∠BCA.
问题：如图所示，∠ABC=30°，∠EAD=55°，AD平分∠EAC。求∠BCA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Shape(EA,AD)
Collinear(BAE)
Collinear(BCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),30)
Equal(MeasureOfAngle(EAD),55)
IsBisectorOfAngle(AD,EAC)
Goal: Value(MeasureOfAngle(BCA))
Answer: 80
Theorem sequence:
angle_addition(1,EAD,DAC)
adjacent_complementary_angle(1,EAC,CAB)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, △ABE is congruent to △DCF, AD=2, BC=2, EF=2, the perimeter of △ABE is 16. Find the perimeter of ABFD.
问题：如图所示，△ABE与△DCF是全等三角形，AD=2，BC=2，EF=2，△ABE的周长为16。求ABFD的周长。
Construction (CDL):
Shape(AB,BC,CG,GA)
Shape(AG,GD,DA)
Shape(GC,CE,EG)
Shape(DG,GE,EF,FD)
Collinear(BCEF)
Collinear(AGE)
Collinear(DGC)
Conditions (CDL):
CongruentBetweenTriangle(ABE,DCF)
Equal(LengthOfLine(AD),2)
Equal(LengthOfLine(BC),2)
Equal(LengthOfLine(EF),2)
Equal(PerimeterOfTriangle(ABE),16)
Goal: Value(PerimeterOfQuadrilateral(ABFD))
Answer: 20
Theorem sequence:
triangle_perimeter_formula(1,ABE)
quadrilateral_perimeter_formula(1,ABFD)
congruent_triangle_property_line_equal(1,BEA,CFD)
line_addition(1,BE,EF)
Problem: As shown in the diagram, ∠ACB=48°, BA is perpendicular to CA. Find the measure of ∠CBA.
问题：如图所示，∠ACB=48°，BA垂直于CA。求∠CBA的大小。
Construction (CDL):
Shape(DC,CA,AE,ED)
Shape(AC,CB,BA)
Shape(EA,AB)
Collinear(CAE)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),48)
PerpendicularBetweenLine(BA,CA)
Goal: Value(MeasureOfAngle(CBA))
Answer: 42
Theorem sequence:
triangle_property_angle_sum(1,BAC)
Problem: As shown in the diagram, △ABC is congruent to △EDF, ∠FED=70°. Find the measure of ∠CAB.
问题：如图所示，三角形ABC与三角形EDF是全等三角形，∠FED=70°。求∠CAB的大小。
Construction (CDL):
Shape(AB,BE,EG,GA)
Shape(GE,ED,DH,HG)
Shape(HD,DC,CH)
Shape(FG,GH,HF)
Shape(AG,GF)
Shape(FH,HC)
Collinear(AGHC)
Collinear(BEDC)
Collinear(FGE)
Collinear(FHD)
Conditions (CDL):
CongruentBetweenTriangle(ABC,EDF)
Equal(MeasureOfAngle(FED),70)
Goal: Value(MeasureOfAngle(CAB))
Answer: 70
Theorem sequence:
congruent_triangle_property_angle_equal(1,ABC,EDF)
Problem: As shown in the diagram, AB=8, AC=8, BC=5, DE is the perpendicular bisector of BA. Find the perimeter of △DCB.
问题：如图所示，AB=8，AC=8，BC=5，DE是BA的垂直平分线。求三角形DCB的周长。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DC,CB,BD)
Shape(ED,DB,BE)
Collinear(ADC)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(BC),5)
IsPerpendicularBisectorOfLine(DE,BA)
Goal: Value(PerimeterOfTriangle(DCB))
Answer: 13
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,DE,BA)
line_addition(1,AD,DC)
triangle_perimeter_formula(1,DCB)
Problem: As shown in the diagram, AB=4, BC=3, DE is parallel to AC, EC∥DB, DABC is a rectangle. Find the perimeter of EDOC.
问题：如图所示，AB=4，BC=3，DE平行于AC，EC∥DB，四边形DABC是长方形。求四边形EDOC的周长。
Construction (CDL):
Shape(DA,AO,OD)
Shape(OA,AB,BO)
Shape(CO,OB,BC)
Shape(DO,OC,CD)
Shape(ED,DC,CE)
Collinear(DOB)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(BC),3)
ParallelBetweenLine(DE,AC)
ParallelBetweenLine(EC,DB)
Rectangle(DABC)
Goal: Value(PerimeterOfQuadrilateral(EDOC))
Answer: 10
Theorem sequence:
right_triangle_judgment_angle(1,ABC)
right_triangle_property_pythagorean(1,ABC)
rectangle_property_diagonal_equal(1,DABC)
parallelogram_property_diagonal_bisection(1,DABC,O)
line_addition(1,DO,OB)
parallelogram_property_diagonal_bisection(1,ABCD,O)
line_addition(1,AO,OC)
parallel_property_collinear_extend(3,CA,ED,O)
parallel_property_collinear_extend(3,BD,CE,O)
parallelogram_judgment_parallel_and_parallel(1,EDOC)
parallelogram_property_opposite_line_equal(1,EDOC)
parallelogram_property_opposite_line_equal(1,DOCE)
quadrilateral_perimeter_formula(1,EDOC)
Problem: As shown in the diagram, ∠DCB=35°, ∠EAB=23°, CD∥AB. Find the measure of ∠AEC.
问题：如图所示，∠DCB=35°，∠EAB=23°，CD平行于AB。求∠AEC的大小。
Construction (CDL):
Shape(DC,CE)
Shape(AE,EC)
Shape(EA,AB,BE)
Collinear(CEB)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),35)
Equal(MeasureOfAngle(EAB),23)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(AEC))
Answer: 58
Theorem sequence:
parallel_property_alternate_interior_angle(1,CD,AB)
triangle_property_angle_sum(1,EAB)
adjacent_complementary_angle(1,BEA,AEC)
Problem: As shown in the diagram, ∠BAC=37°, O is the center of ⊙O. Find the measure of ∠CBO.
问题：如图所示，∠BAC=37°，O是圆O的圆心。求∠CBO的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(OBC,CB)
Shape(DB,BA,AD)
Shape(BD,DC,CB)
Shape(DO,OC,CD)
Shape(OD,DA,OAC,CO)
Collinear(ADC)
Collinear(BDO)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),37)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBO))
Answer: 53
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,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,OCB)
isosceles_triangle_property_angle_equal(1,OCB)
triangle_property_angle_sum(1,OCB)
Problem: As shown in the diagram, BO=CO, ∠COB=100°, BO bisects ∠ABC, CO is the angle bisector of ∠BCA. Find the measure of ∠OAB.
问题：如图所示，BO=CO，∠COB=100°，BO是∠ABC的角平分线，CO平分∠BCA。求∠OAB的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(AO,OC,CA)
Conditions (CDL):
Equal(LengthOfLine(BO),LengthOfLine(CO))
Equal(MeasureOfAngle(COB),100)
IsBisectorOfAngle(BO,ABC)
IsBisectorOfAngle(CO,BCA)
Goal: Value(MeasureOfAngle(OAB))
Answer: 10
Theorem sequence:
incenter_of_triangle_judgment_intersection(1,O,ABC)
triangle_property_angle_sum(1,OBC)
angle_addition(1,ABO,OBC)
angle_addition(1,BCO,OCA)
triangle_property_angle_sum(1,ABC)
angle_addition(1,CAO,OAB)
Problem: As shown in the diagram, AB=EB, DB=CB, EB=DB, ∠EAB=60°, B is the center of circle B, AE∥BC. Find the measure of ∠CDE.
问题：如图所示，AB=EB，DB=CB，EB=DB，∠EAB=60°，B是⊙B的圆心，AE平行于BC。求∠CDE的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BD,DE)
Shape(DB,BC,CD)
Cocircular(B,CDE)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(EB))
Equal(LengthOfLine(DB),LengthOfLine(CB))
Equal(LengthOfLine(EB),LengthOfLine(DB))
Equal(MeasureOfAngle(EAB),60)
IsCentreOfCircle(B,B)
ParallelBetweenLine(AE,BC)
Goal: Value(MeasureOfAngle(CDE))
Answer: 150
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BEA)
isosceles_triangle_property_angle_equal(1,BEA)
parallel_property_alternate_interior_angle(2,AE,BC)
arc_property_center_angle(1,BCE,B)
arc_property_circumference_angle_internal(1,BCE,D)
Problem: As shown in the diagram, AB=AC, BD=CD, ∠DAB=20°, DE is perpendicular to AE. Find the measure of ∠EDC.
问题：如图所示，AB=AC，BD=CD，∠DAB=20°，DE⊥AE。求∠EDC的大小。
Construction (CDL):
Shape(EA,AD,DE)
Shape(CE,ED,DC)
Shape(DA,AB,BD)
Collinear(CEA)
Collinear(CDB)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(BD),LengthOfLine(CD))
Equal(MeasureOfAngle(DAB),20)
PerpendicularBetweenLine(DE,AE)
Goal: Value(MeasureOfAngle(EDC))
Answer: 20
Theorem sequence:
mirror_congruent_triangle_judgment_sss(1,ABD,ADC)
mirror_congruent_triangle_property_angle_equal(1,BDA,CAD)
mirror_congruent_triangle_property_angle_equal(1,DAB,DCA)
adjacent_complementary_angle(1,BDA,ADC)
triangle_property_angle_sum(1,ABD)
adjacent_complementary_angle(1,CED,DEA)
triangle_property_angle_sum(1,CED)
Problem: As shown in the diagram, ∠BND=35°, ∠CAB=30°, quadrilateral DEFG is a rectangle, ∠ABC is the right angle of the right triangle ABC. Find the measure of ∠LMF.
问题：如图所示，∠BND=35°，∠CAB=30°，DEFG是长方形，AB和BC是直角三角形ABC的两个直角边。求∠LMF的大小。
Construction (CDL):
Shape(DE,EB,BN,ND)
Shape(LN,NB,BM,ML)
Shape(AN,NL,LA)
Shape(BC,CM,MB)
Shape(LM,MF,FG,GL)
Collinear(DNLG)
Collinear(EBMF)
Collinear(ANB)
Collinear(ALMC)
Conditions (CDL):
Equal(MeasureOfAngle(BND),35)
Equal(MeasureOfAngle(CAB),30)
Rectangle(DEFG)
RightTriangle(ABC)
Goal: Value(MeasureOfAngle(LMF))
Answer: 65
Theorem sequence:
parallel_property_collinear_extend(3,DG,EF,N)
parallel_property_collinear_extend(3,FE,ND,B)
parallel_property_alternate_interior_angle(2,DN,BF)
angle_addition(1,NBM,MBC)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,BCM)
vertical_angle(1,LMF,CMB)
Problem: As shown in the diagram, AC=7, AC=BC, ∠DCA=90°, AD is the angle bisector of ∠CAB, DE⊥BE. Find the sum of the length of line DE and the length of line BD.
问题：如图所示，AC=7，AC=BC，∠DCA=90°，AD平分∠CAB，DE垂直于BE。求直线DE的长度与直线BD的长度之和。
Construction (CDL):
Shape(CA,AD,DC)
Shape(DA,AE,ED)
Shape(DE,EB,BD)
Collinear(CDB)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AC),7)
Equal(LengthOfLine(AC),LengthOfLine(BC))
Equal(MeasureOfAngle(DCA),90)
IsBisectorOfAngle(AD,CAB)
PerpendicularBetweenLine(DE,BE)
Goal: Value(Add(LengthOfLine(DE),LengthOfLine(BD)))
Answer: 7
Theorem sequence:
adjacent_complementary_angle(1,AED,DEB)
mirror_congruent_triangle_judgment_aas(1,DAE,DCA)
mirror_congruent_triangle_property_line_equal(1,AED,ADC)
line_addition(1,CD,DB)