SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠OAB=35°, the center of ⊙O is O. Find the measure of ∠BCD.
问题：如图所示，∠OAB=35°，O是圆O的圆心。求∠BCD的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(AO,OD,DC,OCA)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 55
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠EDC=40°, BG is parallel to DE, BF is perpendicular to GF. Find the measure of ∠AGF.
问题：如图所示，∠EDC=40°，BG∥DE，BF⊥GF。求∠AGF的大小。
Construction (CDL):
Shape(ED,DC)
Shape(FD,DE)
Shape(AG,GF)
Shape(GF,FD)
Shape(FG,GB,BF)
Collinear(CDFB)
Collinear(AGB)
Conditions (CDL):
Equal(MeasureOfAngle(EDC),40)
ParallelBetweenLine(BG,DE)
PerpendicularBetweenLine(BF,GF)
Goal: Value(MeasureOfAngle(AGF))
Answer: 130
Theorem sequence:
parallel_property_corresponding_angle(2,BG,DE,C)
triangle_property_angle_sum(1,FGB)
adjacent_complementary_angle(1,AGF,FGB)
Problem: As shown in the diagram, ∠CFB=92°, ∠DAE=28°, ∠EBF=∠FCD. Find the measure of ∠CDF.
问题：如图所示，∠CFB=92°，∠DAE=28°，∠EBF=∠FCD。求∠CDF的大小。
Construction (CDL):
Shape(DA,AE,EF,FD)
Shape(FE,EB,BF)
Shape(CD,DF,FC)
Shape(CF,FB)
Collinear(AEB)
Collinear(ADC)
Collinear(CFE)
Collinear(DFB)
Conditions (CDL):
Equal(MeasureOfAngle(CFB),92)
Equal(MeasureOfAngle(DAE),28)
Equal(MeasureOfAngle(EBF),MeasureOfAngle(FCD))
Goal: Value(MeasureOfAngle(CDF))
Answer: 60
Theorem sequence:
adjacent_complementary_angle(1,DFC,CFB)
adjacent_complementary_angle(1,CFB,BFE)
vertical_angle(1,CFB,EFD)
triangle_property_angle_sum(1,CDF)
triangle_property_angle_sum(1,FEB)
quadrilateral_property_angle_sum(1,DAEF)
adjacent_complementary_angle(1,CDF,FDA)
adjacent_complementary_angle(1,AEF,FEB)
Problem: As shown in the diagram, ∠BCD=95°, ∠CDE=25°, GB is parallel to EF. Find the measure of ∠FED.
问题：如图所示，∠BCD=95°，∠CDE=25°，GB∥EF。求∠FED的大小。
Construction (CDL):
Shape(DC,CA)
Shape(CD,DE,EG,GC)
Shape(BG,GE)
Shape(GE,EF)
Shape(FE,ED)
Collinear(ACGB)
Collinear(DEG)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),95)
Equal(MeasureOfAngle(CDE),25)
ParallelBetweenLine(GB,EF)
Goal: Value(MeasureOfAngle(FED))
Answer: 120
Theorem sequence:
triangle_property_angle_sum(1,CDG)
adjacent_complementary_angle(1,BGE,EGC)
parallel_property_corresponding_angle(2,GB,EF,D)
Problem: As shown in the diagram, ∠DAB=110°, E bisects segment AB, F is the midpoint of segment BC, CP is perpendicular to EP, EP is perpendicular to DP, quadrilateral ABCD is a rhombus. Find the measure of ∠CPF.
问题：如图所示，∠DAB=110°，E是线段AB的中点，F是线段BC的中点，CP垂直于EP，EP垂直于DP，ABCD是菱形。求∠CPF的大小。
Construction (CDL):
Shape(AE,EP,PD,DA)
Shape(EF,FP,PE)
Shape(EB,BF,FE)
Shape(PF,FC,CP)
Shape(FB,BG,GF)
Collinear(BFC)
Collinear(PFG)
Collinear(DPC)
Collinear(AEBG)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),110)
IsMidpointOfLine(E,AB)
IsMidpointOfLine(F,BC)
PerpendicularBetweenLine(CP,EP)
PerpendicularBetweenLine(EP,DP)
Rhombus(ABCD)
Goal: Value(MeasureOfAngle(CPF))
Answer: 55
Theorem sequence:
line_addition(1,AE,EB)
line_addition(1,BF,FC)
isosceles_triangle_judgment_line_equal(1,BFE)
isosceles_triangle_property_angle_equal(1,BFE)
parallel_property_ipsilateral_internal_angle(1,AD,BC)
triangle_property_angle_sum(1,BFE)
parallel_property_collinear_extend(1,BA,CD,G)
parallel_property_collinear_extend(2,GB,CD,E)
parallel_property_collinear_extend(3,DC,EB,P)
parallel_property_alternate_interior_angle(2,DC,BG)
vertical_angle(1,PFC,GFB)
congruent_triangle_judgment_aas(2,GFB,PFC)
congruent_triangle_property_line_equal(1,BGF,CPF)
line_addition(1,PF,FG)
median_of_triangle_judgment(1,EF,EGP)
parallel_property_alternate_interior_angle(2,DP,EB)
right_triangle_judgment_angle(1,PEG)
right_triangle_property_length_of_median(1,PEG,F)
isosceles_triangle_judgment_line_equal(1,FEG)
isosceles_triangle_property_angle_equal(1,FEG)
triangle_property_angle_sum(1,PEG)
angle_addition(1,CPF,FPE)
Problem: As shown in the diagram, ∠CEF=110°, ED∥AB. Find the measure of ∠EAB.
问题：如图所示，∠CEF=110°，ED平行于AB。求∠EAB的大小。
Construction (CDL):
Shape(CE,EF)
Shape(FE,ED)
Shape(AE,EC)
Shape(DE,EA)
Shape(EA,AB)
Collinear(CED)
Collinear(FEA)
Conditions (CDL):
Equal(MeasureOfAngle(CEF),110)
ParallelBetweenLine(ED,AB)
Goal: Value(MeasureOfAngle(EAB))
Answer: 70
Theorem sequence:
vertical_angle(1,CEF,DEA)
parallel_property_ipsilateral_internal_angle(1,ED,AB)
Problem: As shown in the diagram, ∠BIE=∠EIG, ∠GIC=40°, IB∥EA, ABCD is a rectangle. Find the measure of ∠IEA.
问题：如图所示，∠BIE=∠EIG，∠GIC=40°，IB∥EA，ABCD是长方形。求∠IEA的大小。
Construction (CDL):
Shape(EA,AB,BI,IE)
Shape(EI,IG,GF,FE)
Shape(HE,EF,FH)
Shape(FG,GI,IC,CD,DF)
Collinear(AEFD)
Collinear(BIC)
Collinear(HFG)
Conditions (CDL):
Equal(MeasureOfAngle(BIE),MeasureOfAngle(EIG))
Equal(MeasureOfAngle(GIC),40)
ParallelBetweenLine(IB,EA)
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(IEA))
Answer: 110
Theorem sequence:
angle_addition(1,BIE,EIG)
angle_addition(1,BIG,GIC)
flat_angle(1,BIC)
parallel_property_ipsilateral_internal_angle(1,IB,EA)
Problem: As shown in the diagram, ∠AQR=∠PQO, ∠QOP=40°, QR∥OP. Find the measure of ∠QPB.
问题：如图所示，∠AQR=∠PQO，∠QOP=40°，QR平行于OP。求∠QPB的大小。
Construction (CDL):
Shape(QO,OP,PQ)
Shape(AQ,QR)
Shape(RQ,QP)
Shape(QP,PB)
Collinear(OPB)
Collinear(AQO)
Conditions (CDL):
Equal(MeasureOfAngle(AQR),MeasureOfAngle(PQO))
Equal(MeasureOfAngle(QOP),40)
ParallelBetweenLine(QR,OP)
Goal: Value(MeasureOfAngle(QPB))
Answer: 80
Theorem sequence:
triangle_property_angle_sum(1,QOP)
parallel_property_corresponding_angle(1,QR,OP,A)
adjacent_complementary_angle(1,OPQ,QPB)
Problem: As shown in the diagram, ∠JIK=30°, ∠JLH=∠CJD, IJ∥KL. Find the measure of ∠IKL.
问题：如图所示，∠JIK=30°，∠JLH=∠CJD，IJ∥KL。求∠IKL的大小。
Construction (CDL):
Shape(AI,IB)
Shape(BI,IJ)
Shape(KI,IA)
Shape(JI,IK)
Shape(EK,KI)
Shape(IK,KE)
Shape(IK,KL)
Shape(LK,KF)
Shape(IJ,JC)
Shape(CJ,JD)
Shape(LJ,JI)
Shape(DJ,JL)
Shape(KL,LJ)
Shape(JL,LH)
Shape(GJ,JK)
Shape(HL,LG)
Collinear(AIJD)
Collinear(EKLH)
Collinear(BIKF)
Collinear(CJLG)
Conditions (CDL):
Equal(MeasureOfAngle(JIK),30)
Equal(MeasureOfAngle(JLH),MeasureOfAngle(CJD))
ParallelBetweenLine(IJ,KL)
Goal: Value(MeasureOfAngle(IKL))
Answer: 150
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,IJ,KL)
Problem: As shown in the diagram, DE=10, D bisects segment AB, E is the midpoint of segment AC, triangle AED is similar to triangle ABC.. Find the length of line BC.
问题：如图所示，DE=10，D平分线段AB，E是线段AC的中点，△AED相似于△ABC。求直线BC的长度。
Construction (CDL):
Shape(DA,AE,ED)
Shape(DE,EC,CB,BD)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(DE),10)
IsMidpointOfLine(D,AB)
IsMidpointOfLine(E,AC)
SimilarBetweenTriangle(AED,ABC)
Goal: Value(LengthOfLine(BC))
Answer: 20
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,ED,ACB)
midsegment_of_triangle_property_length(1,ED,ACB)
Problem: As shown in the diagram, ∠EAB=80°, ∠EBC=40°, triangle ABC is mirror congruent to triangle DBC. Find the measure of ∠ECD.
问题：如图所示，∠EAB=80°，∠EBC=40°，三角形ABC与三角形DBC是镜像全等三角形。求∠ECD的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(DE,EC,CD)
Shape(EB,BC,CE)
Collinear(AEC)
Collinear(BED)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),80)
Equal(MeasureOfAngle(EBC),40)
MirrorCongruentBetweenTriangle(ABC,DBC)
Goal: Value(MeasureOfAngle(ECD))
Answer: 20
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,ABC,DBC)
mirror_congruent_triangle_property_angle_equal(1,CAB,BCD)
triangle_property_angle_sum(1,DBC)
angle_addition(1,BCE,ECD)
Problem: As shown in the diagram, ∠DCF=30°, ∠FEA=45°, AE∥DC, AD⊥FD, FA is perpendicular to BA. Find the measure of ∠FAD.
问题：如图所示，∠DCF=30°，∠FEA=45°，AE∥DC，AD⊥FD，FA⊥BA。求∠FAD的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DF,FA)
Shape(FD,DC,CF)
Shape(AF,FE,EA)
Collinear(AFC)
Collinear(EFD)
Collinear(BDC)
Conditions (CDL):
Equal(MeasureOfAngle(DCF),30)
Equal(MeasureOfAngle(FEA),45)
ParallelBetweenLine(AE,DC)
PerpendicularBetweenLine(AD,FD)
PerpendicularBetweenLine(FA,BA)
Goal: Value(MeasureOfAngle(FAD))
Answer: 15
Theorem sequence:
parallel_property_alternate_interior_angle(1,AE,DC)
triangle_property_angle_sum(1,DEA)
angle_addition(1,EAF,FAD)
Problem: As shown in the diagram, ∠EAD=70°, OD is the perpendicular bisector of AB, OE is the perpendicular bisector of AC. Find the measure of ∠BCO.
问题：如图所示，∠EAD=70°，OD垂直平分AB，OE是AC的垂直平分线。求∠BCO的大小。
Construction (CDL):
Shape(AD,DO,OA)
Shape(AO,OE,EA)
Shape(EO,OC,CE)
Shape(DB,BO,OD)
Shape(OB,BC,CO)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(EAD),70)
IsPerpendicularBisectorOfLine(OD,AB)
IsPerpendicularBisectorOfLine(OE,AC)
Goal: Value(MeasureOfAngle(BCO))
Answer: 20
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,OE,AC)
perpendicular_bisector_property_distance_equal(1,OD,AB)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OCA)
angle_addition(1,CAO,OAD)
quadrilateral_property_angle_sum(1,ABOC)
round_angle(1,COB,BOC)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, ∠CDE=60°, ∠EBA=20°, AB is parallel to EF, EF∥CD. Find the measure of ∠BED.
问题：如图所示，∠CDE=60°，∠EBA=20°，AB平行于EF，EF平行于CD。求∠BED的大小。
Construction (CDL):
Shape(EB,BA)
Shape(BE,EF)
Shape(FE,ED)
Shape(CD,DE)
Conditions (CDL):
Equal(MeasureOfAngle(CDE),60)
Equal(MeasureOfAngle(EBA),20)
ParallelBetweenLine(AB,EF)
ParallelBetweenLine(EF,CD)
Goal: Value(MeasureOfAngle(BED))
Answer: 80
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,EF)
parallel_property_alternate_interior_angle(1,EF,CD)
angle_addition(1,BEF,FED)
Problem: As shown in the diagram, ∠EDA=35°, quadrilateral ABCD is a parallelogram, AE is perpendicular to DE. Find the measure of ∠BCD.
问题：如图所示，∠EDA=35°，BA和CD是平行四边形ABCD的一组对边，AE⊥DE。求∠BCD的大小。
Construction (CDL):
Shape(AE,ED,DA)
Shape(DE,EB,BC,CD)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(EDA),35)
Parallelogram(ABCD)
PerpendicularBetweenLine(AE,DE)
Goal: Value(MeasureOfAngle(BCD))
Answer: 55
Theorem sequence:
triangle_property_angle_sum(1,AED)
parallelogram_property_opposite_angle_equal(1,ABCD)
Problem: As shown in the diagram, ∠CAB=100°, DE is the perpendicular bisector of AB, FG perpendicular bisects AC. Find the measure of ∠FAD.
问题：如图所示，∠CAB=100°，DE是AB的垂直平分线，FG是AC的垂直平分线。求∠FAD的大小。
Construction (CDL):
Shape(BD,DE,EB)
Shape(AE,ED,DA)
Shape(AD,DF,FA)
Shape(AF,FG,GA)
Shape(GF,FC,CG)
Collinear(BEA)
Collinear(AGC)
Collinear(BDFC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),100)
IsPerpendicularBisectorOfLine(DE,AB)
IsPerpendicularBisectorOfLine(FG,AC)
Goal: Value(MeasureOfAngle(FAD))
Answer: 20
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,DE,AB)
perpendicular_bisector_property_distance_equal(1,FG,AC)
isosceles_triangle_judgment_line_equal(1,DAB)
isosceles_triangle_judgment_line_equal(1,FCA)
isosceles_triangle_property_angle_equal(1,DAB)
isosceles_triangle_property_angle_equal(1,FCA)
triangle_property_angle_sum(1,ABC)
angle_addition(1,GAF,FAD)
angle_addition(1,GAD,DAE)
Problem: As shown in the diagram, ∠CAB=70°, BD is the angle bisector of ∠CBE, CD bisects ∠FCB. Find the measure of ∠BDC.
问题：如图所示，∠CAB=70°，BD平分∠CBE，CD是∠FCB的角平分线。求∠BDC的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BD,DC,CB)
Shape(DB,BE)
Shape(FC,CD)
Collinear(ABE)
Collinear(ACF)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),70)
IsBisectorOfAngle(BD,CBE)
IsBisectorOfAngle(CD,FCB)
Goal: Value(MeasureOfAngle(BDC))
Answer: 55
Theorem sequence:
angle_addition(1,ABC,CBE)
angle_addition(1,CBD,DBE)
angle_addition(1,FCD,DCB)
angle_addition(1,FCB,BCA)
flat_angle(1,ABE)
flat_angle(1,FCA)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,BDC)
Problem: As shown in the diagram, ∠ACE=40°, AE bisects ∠BAC, AB∥CE. Find the measure of ∠CEA.
问题：如图所示，∠ACE=40°，AE是∠BAC的角平分线，AB平行于CE。求∠CEA的大小。
Construction (CDL):
Shape(AC,CE,EA)
Shape(DA,AB)
Shape(BA,AE)
Collinear(DAC)
Conditions (CDL):
Equal(MeasureOfAngle(ACE),40)
IsBisectorOfAngle(AE,BAC)
ParallelBetweenLine(AB,CE)
Goal: Value(MeasureOfAngle(CEA))
Answer: 70
Theorem sequence:
parallel_property_corresponding_angle(1,AB,CE,D)
angle_addition(1,DAB,BAE)
angle_addition(1,DAE,EAC)
flat_angle(1,DAC)
triangle_property_angle_sum(1,ACE)
Problem: As shown in the diagram, ∠BDA=35°, ∠DAB=40°, BD∥CE. Find the measure of ∠BCE.
问题：如图所示，∠BDA=35°，∠DAB=40°，BD∥CE。求∠BCE的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DB,BC)
Shape(BC,CE)
Collinear(ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BDA),35)
Equal(MeasureOfAngle(DAB),40)
ParallelBetweenLine(BD,CE)
Goal: Value(MeasureOfAngle(BCE))
Answer: 105
Theorem sequence:
triangle_property_angle_sum(1,ABD)
parallel_property_corresponding_angle(1,BD,CE,A)
Problem: As shown in the diagram, ∠AOC=130°, the center of ⊙O is O. Find the measure of ∠EDB.
问题：如图所示，∠AOC=130°，O是圆O的圆心。求∠EDB的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(CE,EB,OBC)
Shape(CO,OE,EC)
Shape(AO,OC,OCA)
Shape(DB,BE,ED)
Shape(DE,EO,OA,OAD)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,BCAD)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(EDB))
Answer: 25
Theorem sequence:
adjacent_complementary_angle(1,AOC,COE)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠CDO=80°, ∠DOC=70°, △OAB is mirror congruent to △OCD. Find the measure of ∠ABO.
问题：如图所示，∠CDO=80°，∠DOC=70°，三角形OAB镜像全等于三角形OCD。求∠ABO的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(DO,OC,CD)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(MeasureOfAngle(CDO),80)
Equal(MeasureOfAngle(DOC),70)
MirrorCongruentBetweenTriangle(OAB,OCD)
Goal: Value(MeasureOfAngle(ABO))
Answer: 30
Theorem sequence:
triangle_property_angle_sum(1,DOC)
mirror_congruent_triangle_property_angle_equal(1,BOA,CDO)
Problem: As shown in the diagram, ∠ABE=35°, ∠ECD=50°, BF bisects ∠ABE, BF is perpendicular to AF. Find the measure of ∠CDE.
问题：如图所示，∠ABE=35°，∠ECD=50°，BF平分∠ABE，BF⊥AF。求∠CDE的大小。
Construction (CDL):
Shape(AB,BF,FA)
Shape(FB,BE,EF)
Shape(AF,FD,DA)
Shape(FE,ED,DF)
Shape(DE,EC,CD)
Collinear(BFD)
Collinear(AFE)
Collinear(ADC)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),35)
Equal(MeasureOfAngle(ECD),50)
IsBisectorOfAngle(BF,ABE)
PerpendicularBetweenLine(BF,AF)
Goal: Value(MeasureOfAngle(CDE))
Answer: 45
Theorem sequence:
angle_addition(1,ABF,FBE)
angle_addition(1,DAF,FAB)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,ABF)
adjacent_complementary_angle(1,EFB,BFA)
adjacent_complementary_angle(1,BFA,AFD)
vertical_angle(1,BFA,DFE)
mirror_congruent_triangle_judgment_aas(2,ABF,EFB)
mirror_congruent_triangle_property_line_equal(1,BFA,BEF)
mirror_congruent_triangle_judgment_sas(1,FDA,FED)
mirror_congruent_triangle_property_angle_equal(1,AFD,EDF)
triangle_property_angle_sum(1,AFD)
triangle_property_angle_sum(1,DFE)
angle_addition(1,EDF,FDA)
adjacent_complementary_angle(1,CDE,EDA)
Problem: As shown in the diagram, ∠BCA=60°, AD is parallel to BC, CA⊥BA. Find the measure of ∠EAD.
问题：如图所示，∠BCA=60°，AD平行于BC，CA⊥BA。求∠EAD的大小。
Construction (CDL):
Shape(EA,AD)
Shape(DA,AC)
Shape(AB,BC,CA)
Collinear(EAB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),60)
ParallelBetweenLine(AD,BC)
PerpendicularBetweenLine(CA,BA)
Goal: Value(MeasureOfAngle(EAD))
Answer: 30
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,BC)
adjacent_complementary_angle(1,EAC,CAB)
angle_addition(1,EAD,DAC)
Problem: As shown in the diagram, Add(MeasureOfAngle(CBD)=MeasureOfAngle(BDE)), ∠JIH=55°. Find the measure of ∠DKL.
问题：如图所示，Add(MeasureOfAngle(CBD)=MeasureOfAngle(BDE))，∠JIH=55°。求∠DKL的大小。
Construction (CDL):
Shape(AB,BC)
Shape(CB,BD)
Shape(IB,BA)
Shape(DB,BI)
Shape(BD,DE)
Shape(ED,DF)
Shape(FD,DK)
Shape(KD,DB)
Shape(HI,IB)
Shape(BI,IK)
Shape(KI,IJ)
Shape(JI,JH)
Shape(IK,KD)
Shape(DK,KL)
Shape(LK,KM)
Shape(MK,KI)
Collinear(ABDF)
Collinear(CBIJ)
Collinear(HIKL)
Collinear(EDKM)
Conditions (CDL):
Equal(Add(MeasureOfAngle(CBD),MeasureOfAngle(BDE)),180)
Equal(MeasureOfAngle(JIH),55)
Goal: Value(MeasureOfAngle(DKL))
Answer: 55
Theorem sequence:
parallel_judgment_ipsilateral_internal_angle(1,BC,DE)
parallel_property_collinear_extend(1,BC,DE,I)
parallel_property_collinear_extend(2,ED,BI,K)
vertical_angle(1,JIH,BIK)
parallel_property_corresponding_angle(2,IB,KE,L)
Problem: As shown in the diagram, ∠ECD=70°, ∠EFA=30°, AB∥CD. Find the measure of ∠FAE.
问题：如图所示，∠ECD=70°，∠EFA=30°，AB平行于CD。求∠FAE的大小。
Construction (CDL):
Shape(FA,AE,EF)
Shape(FE,EB)
Shape(EC,CD)
Shape(CE,EA)
Collinear(AEB)
Collinear(FEC)
Conditions (CDL):
Equal(MeasureOfAngle(ECD),70)
Equal(MeasureOfAngle(EFA),30)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(FAE))
Answer: 40
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,E)
parallel_property_corresponding_angle(1,EB,CD,F)
adjacent_complementary_angle(1,AEF,FEB)
triangle_property_angle_sum(1,FAE)
Problem: As shown in the diagram, ∠DEA=60°, DE is parallel to FG, EA⊥DA. Find the measure of ∠GFB.
问题：如图所示，∠DEA=60°，DE平行于FG，EA⊥DA。求∠GFB的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DF,FG,GE,ED)
Shape(FB,BC,CG,GF)
Collinear(ADFB)
Collinear(AEGC)
Conditions (CDL):
Equal(MeasureOfAngle(DEA),60)
ParallelBetweenLine(DE,FG)
PerpendicularBetweenLine(EA,DA)
Goal: Value(MeasureOfAngle(GFB))
Answer: 150
Theorem sequence:
triangle_property_angle_sum(1,ADE)
parallel_property_corresponding_angle(1,DE,FG,A)
adjacent_complementary_angle(1,AFG,GFB)
Problem: As shown in the diagram, AB=5, AC=3, AC⊥BC. Find the length of line BC.
问题：如图所示，AB=5，AC=3，AC⊥BC。求直线BC的长度。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(AC),3)
PerpendicularBetweenLine(AC,BC)
Goal: Value(LengthOfLine(BC))
Answer: 4
Theorem sequence:
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
Problem: As shown in the diagram, ∠AEG=110°, AB∥CD. Find the measure of ∠EFD.
问题：如图所示，∠AEG=110°，AB∥CD。求∠EFD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,DH)
Shape(HF,FC)
Collinear(AEB)
Collinear(CFD)
Collinear(GEFH)
Conditions (CDL):
Equal(MeasureOfAngle(AEG),110)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(EFD))
Answer: 70
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,E)
parallel_property_collinear_extend(3,DC,EA,F)
parallel_property_corresponding_angle(2,FC,EA,G)
adjacent_complementary_angle(1,CFE,EFD)
Problem: As shown in the diagram, ∠ADE=105°, ∠ECF=140°, D bisects segment AB, E bisects segment AC. Find the measure of ∠EAD.
问题：如图所示，∠ADE=105°，∠ECF=140°，D平分线段AB，E是线段AC的中点。求∠EAD的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Shape(EC,CF)
Collinear(ADB)
Collinear(AEC)
Collinear(BCF)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),105)
Equal(MeasureOfAngle(ECF),140)
IsMidpointOfLine(D,AB)
IsMidpointOfLine(E,AC)
Goal: Value(MeasureOfAngle(EAD))
Answer: 35
Theorem sequence:
adjacent_complementary_angle(1,BCE,ECF)
midsegment_of_triangle_judgment_midpoint(1,DE,ABC)
midsegment_of_triangle_property_parallel(1,DE,ABC)
parallel_property_corresponding_angle(2,CB,ED,A)
triangle_property_angle_sum(1,ADE)
Problem: As shown in the diagram, ∠COB=42°, AO is perpendicular to BO. Find the measure of ∠AOC.
问题：如图所示，∠COB=42°，AO垂直于BO。求∠AOC的大小。
Construction (CDL):
Shape(AO,OC)
Shape(CO,OB)
Conditions (CDL):
Equal(MeasureOfAngle(COB),42)
PerpendicularBetweenLine(AO,BO)
Goal: Value(MeasureOfAngle(AOC))
Answer: 48
Theorem sequence:
angle_addition(1,AOC,COB)
Problem: As shown in the diagram, ∠BCA=42°, ∠CAB=78°, AD is parallel to BC. Find the measure of ∠BAE.
问题：如图所示，∠BCA=42°，∠CAB=78°，AD平行于BC。求∠BAE的大小。
Construction (CDL):
Shape(BA,AE)
Shape(DA,AC)
Shape(AB,BC,CA)
Collinear(EAD)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),42)
Equal(MeasureOfAngle(CAB),78)
ParallelBetweenLine(AD,BC)
Goal: Value(MeasureOfAngle(BAE))
Answer: 60
Theorem sequence:
triangle_property_angle_sum(1,ABC)
parallel_property_collinear_extend(1,AD,BC,E)
parallel_property_alternate_interior_angle(2,EA,BC)
Problem: As shown in the diagram, the perimeter of quadrilateral ABCD is 16, BA and CD are opposite sides of the ▱ ABCD, EO⊥CO. Find the perimeter of △DEC.
问题：如图所示，ABCD的周长为16，四边形ABCD是▱，EO垂直于CO。求三角形DEC的周长。
Construction (CDL):
Shape(EA,AO,OE)
Shape(FE,EO,OF)
Shape(DE,EF,FD)
Shape(DF,FC,CD)
Shape(CF,FO,OC)
Shape(CO,OB,BC)
Shape(BO,OA,AB)
Collinear(AOC)
Collinear(AED)
Collinear(EFC)
Collinear(BOFD)
Conditions (CDL):
Equal(PerimeterOfQuadrilateral(ABCD),16)
Parallelogram(ABCD)
PerpendicularBetweenLine(EO,CO)
Goal: Value(PerimeterOfTriangle(DEC))
Answer: 8
Theorem sequence:
quadrilateral_perimeter_formula(1,ABCD)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
parallelogram_property_diagonal_bisection(1,ABCD,O)
adjacent_complementary_angle(1,AOE,EOC)
perpendicular_bisector_judgment_per_and_mid(1,EO,AC)
perpendicular_bisector_property_distance_equal(1,EO,AC)
line_addition(1,AE,ED)
triangle_perimeter_formula(1,DEC)
Problem: As shown in the diagram, ∠ABC=70°, the center of ⊙O is O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=70°，圆O的圆心为O。求∠AOC的大小。
Construction (CDL):
Shape(AB,BC,CO,OA)
Shape(AO,OC,CA)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCA,AC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠CEF=45°, ∠ECD=20°, AB∥CD. Find the measure of ∠AEF.
问题：如图所示，∠CEF=45°，∠ECD=20°，AB∥CD。求∠AEF的大小。
Construction (CDL):
Shape(EA,AC,CE)
Shape(BE,EC,CD,DB)
Shape(FE,EB)
Shape(AE,EF)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(CEF),45)
Equal(MeasureOfAngle(ECD),20)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(AEF))
Answer: 25
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,E)
parallel_property_alternate_interior_angle(2,AE,CD)
angle_addition(1,CEA,AEF)
Problem: As shown in the diagram, MO=5, M bisects segment AB, quadrilateral ABCD is a rhombus. Find the perimeter of quadrilateral ABCD.
问题：如图所示，MO=5，M是线段AB的中点，ABCD是菱形。求四边形ABCD的周长。
Construction (CDL):
Shape(AM,MO,OA)
Shape(MB,BO,OM)
Shape(OB,BC,CO)
Shape(DO,OC,CD)
Shape(OD,DA,AO)
Collinear(AOC)
Collinear(BOD)
Collinear(AMB)
Conditions (CDL):
Equal(LengthOfLine(MO),5)
IsMidpointOfLine(M,AB)
Rhombus(ABCD)
Goal: Value(PerimeterOfQuadrilateral(ABCD))
Answer: 40
Theorem sequence:
parallelogram_property_diagonal_bisection(1,ABCD,O)
midsegment_of_triangle_judgment_midpoint(1,MO,ABC)
midsegment_of_triangle_property_length(1,MO,ABC)
quadrilateral_perimeter_formula(1,ABCD)
Problem: As shown in the diagram, ∠CAD=30°, ∠HED=50°, GC∥HF. Find the measure of ∠DCA.
问题：如图所示，∠CAD=30°，∠HED=50°，GC∥HF。求∠DCA的大小。
Construction (CDL):
Shape(AD,DC,CA)
Shape(DE,EF,FC,CD)
Shape(EB,BF,FE)
Shape(GD,DA)
Shape(ED,DG)
Shape(HE,ED)
Shape(BE,EH)
Collinear(ADEB)
Collinear(GDC)
Collinear(HEF)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),30)
Equal(MeasureOfAngle(HED),50)
ParallelBetweenLine(GC,HF)
Goal: Value(MeasureOfAngle(DCA))
Answer: 20
Theorem sequence:
parallel_property_collinear_extend(3,GC,HF,D)
parallel_property_collinear_extend(3,FH,DG,E)
parallel_property_corresponding_angle(2,EH,DG,A)
adjacent_complementary_angle(1,GDA,ADC)
triangle_property_angle_sum(1,ADC)
Problem: As shown in the diagram, ∠FAB=115°, CD∥AB. Find the measure of ∠CFE.
问题：如图所示，∠FAB=115°，CD∥AB。求∠CFE的大小。
Construction (CDL):
Shape(FA,AB)
Shape(DF,FA)
Shape(EF,FD)
Shape(CF,FE)
Shape(AF,FC)
Collinear(AFE)
Collinear(CFD)
Conditions (CDL):
Equal(MeasureOfAngle(FAB),115)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(CFE))
Answer: 65
Theorem sequence:
parallel_property_collinear_extend(3,CD,AB,F)
parallel_property_corresponding_angle(1,FD,AB,E)
adjacent_complementary_angle(1,CFE,EFD)