SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BCM=30°, ∠HAN=45°, DEFG is a rectangle, ∠BCA is the right angle of the right △ BCA. Find the measure of ∠NHA.
问题：如图所示，∠BCM=30°，∠HAN=45°，四边形DEFG是长方形，BC和CA是直角△BCA的两个直角边。求∠NHA的大小。
Construction (CDL):
Shape(DE,EM,MN,ND)
Shape(AN,NH,HA)
Shape(NM,MC,CH,HN)
Shape(MB,BC,CM)
Shape(HC,CF,FG,GH)
Collinear(DNHG)
Collinear(EMCF)
Collinear(ANMB)
Collinear(AHC)
Conditions (CDL):
Equal(MeasureOfAngle(BCM),30)
Equal(MeasureOfAngle(HAN),45)
Rectangle(DEFG)
RightTriangle(BCA)
Goal: Value(MeasureOfAngle(NHA))
Answer: 60
Theorem sequence:
parallel_property_collinear_extend(3,DG,EF,H)
parallel_property_collinear_extend(3,FE,HD,C)
parallel_property_corresponding_angle(2,CE,HD,A)
angle_addition(1,BCM,MCH)
Problem: As shown in the diagram, ∠ABO=40°, ∠ODC=30°, BA∥DC. Find the measure of ∠AOC.
问题：如图所示，∠ABO=40°，∠ODC=30°，BA∥DC。求∠AOC的大小。
Construction (CDL):
Shape(CO,OD,DC)
Shape(OA,AB,BO)
Shape(AO,OC)
Shape(DO,OB)
Collinear(COB)
Collinear(DOA)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),40)
Equal(MeasureOfAngle(ODC),30)
ParallelBetweenLine(BA,DC)
Goal: Value(MeasureOfAngle(AOC))
Answer: 70
Theorem sequence:
parallel_property_alternate_interior_angle(1,BA,DC)
triangle_property_angle_sum(1,COD)
adjacent_complementary_angle(1,AOC,COD)
Problem: As shown in the diagram, AC=2, CB=8, CD=8/5, AC is perpendicular to DC, CB is perpendicular to EB. Find the length of line EB.
问题：如图所示，AC=2，CB=8，CD=8/5，AC⊥DC，CB垂直于EB。求直线EB的长度。
Construction (CDL):
Shape(AC,CD,DA)
Shape(ED,DC,CB,BE)
Collinear(ADE)
Collinear(ACB)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(LengthOfLine(CB),8)
Equal(LengthOfLine(CD),8/5)
PerpendicularBetweenLine(AC,DC)
PerpendicularBetweenLine(CB,EB)
Goal: Value(LengthOfLine(EB))
Answer: 8
Theorem sequence:
line_addition(1,AC,CB)
similar_triangle_judgment_aa(1,DAC,EAB)
similar_triangle_property_line_ratio(1,DAC,EAB)
similar_triangle_property_line_ratio(1,ACD,ABE)
Problem: As shown in the diagram, AD=2, DB=1, DE=3/2, DE∥BC. Find the length of line BC.
问题：如图所示，AD=2，DB=1，DE=3/2，DE平行于BC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),2)
Equal(LengthOfLine(DB),1)
Equal(LengthOfLine(DE),3/2)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 9/4
Theorem sequence:
line_addition(1,AD,DB)
parallel_property_corresponding_angle(1,DE,BC,A)
parallel_property_corresponding_angle(2,CB,ED,A)
similar_triangle_judgment_aa(1,ADE,ABC)
similar_triangle_property_line_ratio(1,ADE,ABC)
similar_triangle_property_line_ratio(1,EAD,CAB)
Problem: As shown in the diagram, BC=3, DE is the midsegment of triangle ABC. Find the length of line DE.
问题：如图所示，BC=3，DE是三角形ABC的中位线。求直线DE的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(BC),3)
IsMidsegmentOfTriangle(DE,ABC)
Goal: Value(LengthOfLine(DE))
Answer: 3/2
Theorem sequence:
midsegment_of_triangle_property_length(1,DE,ABC)
Problem: As shown in the diagram, ∠CAN=30°, ∠NAE=75°, CP is the angle bisector of ∠DCA, AM is perpendicular to EM, EF⊥DF, EN is perpendicular to AN. Find the measure of ∠PED.
问题：如图所示，∠CAN=30°，∠NAE=75°，CP是∠DCA的角平分线，AM⊥EM，EF垂直于DF，EN⊥AN。求∠PED的大小。
Construction (CDL):
Shape(AM,ME,EA)
Shape(AE,EN,NA)
Shape(NE,EP,PN)
Shape(AN,NP,PC,CA)
Shape(PE,ED,DP)
Shape(EB,BF,FE)
Shape(EF,FD,DE)
Shape(PD,DC,CP)
Collinear(MAC)
Collinear(EPC)
Collinear(BFDC)
Collinear(ANPD)
Conditions (CDL):
Equal(MeasureOfAngle(CAN),30)
Equal(MeasureOfAngle(NAE),75)
IsBisectorOfAngle(CP,DCA)
PerpendicularBetweenLine(AM,EM)
PerpendicularBetweenLine(EF,DF)
PerpendicularBetweenLine(EN,AN)
Goal: Value(MeasureOfAngle(PED))
Answer: 15
Theorem sequence:
angle_addition(1,CAN,NAE)
adjacent_complementary_angle(1,CAE,EAM)
mirror_congruent_triangle_judgment_aas(1,EAM,ENA)
mirror_congruent_triangle_judgment_aas(1,ECM,EFC)
mirror_congruent_triangle_property_line_equal(1,AME,AEN)
mirror_congruent_triangle_property_line_equal(1,CME,CEF)
adjacent_complementary_angle(1,PNE,ENA)
mirror_congruent_triangle_judgment_hl(1,DNE,DEF)
mirror_congruent_triangle_property_angle_equal(1,DNE,DEF)
angle_addition(1,FDE,EDP)
adjacent_complementary_angle(1,FDE,EDC)
triangle_property_angle_sum(1,EDC)
angle_addition(1,DCP,PCA)
adjacent_complementary_angle(1,FDP,PDC)
triangle_property_angle_sum(1,ADC)
Problem: As shown in the diagram, CD=2, MC=3, ND=3, ∠DEN=45°, AB⊥CB, MC⊥DC, ND⊥ED. Find the length of line AB.
问题：如图所示，CD=2，MC=3，ND=3，∠DEN=45°，AB垂直于CB，MC垂直于DC，ND垂直于ED。求直线AB的长度。
Construction (CDL):
Shape(AB,BC,CM,MA)
Shape(AM,MD,DN,NA)
Shape(MC,CD,DM)
Shape(ND,DE,EN)
Collinear(BCDE)
Collinear(ANE)
Collinear(AMD)
Conditions (CDL):
Equal(LengthOfLine(CD),2)
Equal(LengthOfLine(MC),3)
Equal(LengthOfLine(ND),3)
Equal(MeasureOfAngle(DEN),45)
PerpendicularBetweenLine(AB,CB)
PerpendicularBetweenLine(MC,DC)
PerpendicularBetweenLine(ND,ED)
Goal: Value(LengthOfLine(AB))
Answer: 9
Theorem sequence:
triangle_property_angle_sum(1,ABE)
isosceles_triangle_judgment_angle_equal(1,BEA)
line_addition(1,BD,DE)
similar_triangle_judgment_aa(1,MCD,ABD)
similar_triangle_judgment_aa(1,NDE,ABE)
similar_triangle_property_line_ratio(1,MCD,ABD)
similar_triangle_property_line_ratio(1,DMC,DAB)
similar_triangle_property_line_ratio(1,NDE,ABE)
similar_triangle_property_line_ratio(1,END,EAB)
Problem: As shown in the diagram, AB=AC, ∠ABD=40°, D is the midpoint of segment BC. Find the measure of ∠DAB.
问题：如图所示，AB=AC，∠ABD=40°，D平分线段BC。求∠DAB的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(ABD),40)
IsMidpointOfLine(D,BC)
Goal: Value(MeasureOfAngle(DAB))
Answer: 50
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
mirror_congruent_triangle_judgment_sas(1,BDA,CAD)
mirror_congruent_triangle_property_angle_equal(1,DAB,DCA)
adjacent_complementary_angle(1,BDA,ADC)
triangle_property_angle_sum(1,ABD)
Problem: As shown in the diagram, AB=5, BC=3, △ACB is a right △. Find the sum of the length of line AC and the length of line BC.
问题：如图所示，AB=5，BC=3，△ACB为直角△。求直线AC的长度与直线BC的长度之和。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(BC),3)
RightTriangle(ACB)
Goal: Value(Add(LengthOfLine(AC),LengthOfLine(BC)))
Answer: 7
Theorem sequence:
right_triangle_property_pythagorean(1,ACB)
Problem: As shown in the diagram, ∠ABH=20°, ∠HAB=30°, quadrilateral DEFG is a rectangle, AB and BC are the legs of the right triangle ABC. Find the measure of ∠HCF.
问题：如图所示，∠ABH=20°，∠HAB=30°，DEFG是矩形，∠ABC是直角△ABC的直角。求∠HCF的大小。
Construction (CDL):
Shape(AB,BH,HA)
Shape(BD,DE,EC,CB)
Shape(HB,BC,CH)
Shape(GH,HC,CF,FG)
Collinear(DBHG)
Collinear(ECF)
Collinear(CHA)
Conditions (CDL):
Equal(MeasureOfAngle(ABH),20)
Equal(MeasureOfAngle(HAB),30)
Rectangle(DEFG)
RightTriangle(ABC)
Goal: Value(MeasureOfAngle(HCF))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,ABH)
adjacent_complementary_angle(1,CHB,BHA)
parallel_property_collinear_extend(3,DG,EF,H)
parallel_property_collinear_extend(3,FE,HD,C)
parallel_property_alternate_interior_angle(2,DH,CF)
Problem: As shown in the diagram, ∠ABC=65°, ∠CAB=40°. Find the measure of ∠BCA.
问题：如图所示，∠ABC=65°，∠CAB=40°。求∠BCA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),65)
Equal(MeasureOfAngle(CAB),40)
Goal: Value(MeasureOfAngle(BCA))
Answer: 75
Theorem sequence:
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, AB=6, AC=7, BC=5, DE is the perpendicular bisector of AB. Find the perimeter of △BCD.
问题：如图所示，AB=6，AC=7，BC=5，DE是AB的垂直平分线。求△BCD的周长。
Construction (CDL):
Shape(AE,ED,DA)
Shape(DE,EB,BD)
Shape(DB,BC,CD)
Collinear(ADC)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(AC),7)
Equal(LengthOfLine(BC),5)
IsPerpendicularBisectorOfLine(DE,AB)
Goal: Value(PerimeterOfTriangle(BCD))
Answer: 12
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,DE,AB)
line_addition(1,AD,DC)
triangle_perimeter_formula(1,BCD)
Problem: As shown in the diagram, ∠CAO=20°, O is the center of ⊙O, OE⊥CE. Find the measure of ∠EOD.
问题：如图所示，∠CAO=20°，O是圆O的圆心，OE⊥CE。求∠EOD的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OBC,CE,EB)
Shape(EC,CA,AO,OE)
Shape(OAD,DO,OA)
Shape(ODB,BE,ED)
Shape(OD,DE,EO)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),20)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,CE)
Goal: Value(MeasureOfAngle(EOD))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
adjacent_complementary_angle(1,DEO,OEC)
circle_property_chord_perpendicular_bisect_arc(1,ODC,OEB)
congruent_arc_judgment_length_equal(1,ODB,OBC)
congruent_arc_property_measure_equal(1,ODB,OBC)
arc_property_center_angle(1,ODB,O)
Problem: As shown in the diagram, ∠DBO=63°, the center of circle O is O. Find the measure of ∠BCE.
问题：如图所示，∠DBO=63°，O是圆O的圆心。求∠BCE的大小。
Construction (CDL):
Shape(OAD,DE,EA)
Shape(ODB,BD)
Shape(ED,DO,OE)
Shape(OD,DB,BO)
Shape(OBC,CB)
Shape(OCA,AE,EC)
Shape(CE,EO,OB,BC)
Collinear(BOEA)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(DBO),63)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCE))
Answer: 27
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,ODB,C)
arc_property_center_angle(1,OAD,O)
arc_property_center_angle(1,ODB,O)
adjacent_complementary_angle(1,BOD,DOE)
Problem: As shown in the diagram, ∠ACE=130°, ∠CAB=70°. Find the measure of ∠DBA.
问题：如图所示，∠ACE=130°，∠CAB=70°。求∠DBA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DB,BA)
Shape(AC,CE)
Collinear(DBCE)
Conditions (CDL):
Equal(MeasureOfAngle(ACE),130)
Equal(MeasureOfAngle(CAB),70)
Goal: Value(MeasureOfAngle(DBA))
Answer: 120
Theorem sequence:
adjacent_complementary_angle(1,DBA,ABC)
adjacent_complementary_angle(1,DCA,ACE)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠NAM=110°, DM is the perpendicular bisector of AB, EN is the perpendicular bisector of AC. Find the measure of ∠EAD.
问题：如图所示，∠NAM=110°，DM是AB的垂直平分线，EN垂直平分AB。求∠EAD的大小。
Construction (CDL):
Shape(AM,MD,DA)
Shape(MB,BD,DM)
Shape(AD,DE,EA)
Shape(AE,EN,NA)
Shape(NE,EC,CN)
Collinear(AMB)
Collinear(ANC)
Collinear(BDEC)
Conditions (CDL):
Equal(MeasureOfAngle(NAM),110)
IsPerpendicularBisectorOfLine(DM,AB)
IsPerpendicularBisectorOfLine(EN,AC)
Goal: Value(MeasureOfAngle(EAD))
Answer: 40
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,DM,AB)
perpendicular_bisector_property_distance_equal(1,EN,AC)
isosceles_triangle_judgment_line_equal(1,DAB)
isosceles_triangle_judgment_line_equal(1,ECA)
isosceles_triangle_property_angle_equal(1,DAB)
isosceles_triangle_property_angle_equal(1,ECA)
triangle_property_angle_sum(1,ABC)
angle_addition(1,EAD,DAM)
angle_addition(1,NAE,EAM)
Problem: As shown in the diagram, ∠CDA=70°, AD is the angle bisector of ∠BAC, AB∥CD. Find the measure of ∠DAC.
问题：如图所示，∠CDA=70°，AD是∠BAC的角平分线，AB平行于CD。求∠DAC的大小。
Construction (CDL):
Shape(AC,CD,DA)
Shape(BA,AD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),70)
IsBisectorOfAngle(AD,BAC)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(DAC))
Answer: 70
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,CD)
Problem: As shown in the diagram, ∠EPF=35°, O is the center of circle O, AD is the tangent to ⊙O. Find the measure of ∠DAE.
问题：如图所示，∠EPF=35°，O是圆O的圆心，圆O的切线为AD。求∠DAE的大小。
Construction (CDL):
Shape(ODE,DA,AE)
Shape(ODE,EF,FD)
Shape(OD,DF,FO)
Shape(OCD,DO,OC)
Shape(OEP,PE)
Shape(FE,EP,PF)
Shape(OPC,CO,OF,FP)
Collinear(AEFOC)
Collinear(ADB)
Collinear(DFP)
Cocircular(O,CDEP)
Conditions (CDL):
Equal(MeasureOfAngle(EPF),35)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AD,O)
Goal: Value(MeasureOfAngle(DAE))
Answer: 20
Theorem sequence:
arc_property_circumference_angle_external(1,ODE,P)
arc_property_center_angle(1,OCD,O)
arc_property_center_angle(1,ODE,O)
adjacent_complementary_angle(1,FOD,DOC)
circle_property_circular_power_tangent_and_segment_angle(1,AD,AEC,O)
Problem: As shown in the diagram, ∠DAB=120°, MB perpendicular bisects AP, ND perpendicular bisects QA. Find the sum of the measure of ∠AMN and the measure of ∠MNA.
问题：如图所示，∠DAB=120°，MB是AP的垂直平分线，ND是QA的垂直平分线。求∠AMN的大小与∠MNA的大小之和。
Construction (CDL):
Shape(AB,BM,MA)
Shape(BP,PM,MB)
Shape(AM,MN,NA)
Shape(DA,AN,ND)
Shape(NM,MC,CN)
Shape(QD,DN,NQ)
Collinear(ABP)
Collinear(ADQ)
Collinear(PMNQ)
Collinear(BMC)
Collinear(DNC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),120)
IsPerpendicularBisectorOfLine(MB,AP)
IsPerpendicularBisectorOfLine(ND,QA)
Goal: Value(Add(MeasureOfAngle(AMN),MeasureOfAngle(MNA)))
Answer: 120
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,MB,AP)
perpendicular_bisector_property_distance_equal(1,ND,QA)
isosceles_triangle_judgment_line_equal(1,MAP)
isosceles_triangle_judgment_line_equal(1,NQA)
isosceles_triangle_property_angle_equal(1,MAP)
isosceles_triangle_property_angle_equal(1,NQA)
triangle_property_angle_sum(1,APM)
adjacent_complementary_angle(1,PMA,AMN)
triangle_property_angle_sum(1,ANQ)
adjacent_complementary_angle(1,MNA,ANQ)
triangle_property_angle_sum(1,AMN)
angle_addition(1,DAN,NAM)
angle_addition(1,DAM,MAB)
Problem: As shown in the diagram, Add(MeasureOfAngle(BCA)=MeasureOfAngle(CAB)), AB=AC, AD=AE, EA is perpendicular to DA. Find the measure of ∠EDC.
问题：如图所示，Add(MeasureOfAngle(BCA)=MeasureOfAngle(CAB))，AB=AC，AD=AE，EA⊥DA。求∠EDC的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Collinear(BDC)
Collinear(AEC)
Conditions (CDL):
Equal(Add(MeasureOfAngle(BCA),MeasureOfAngle(CAB)),145)
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(AD),LengthOfLine(AE))
PerpendicularBetweenLine(EA,DA)
Goal: Value(MeasureOfAngle(EDC))
Answer: 10
Theorem sequence:
triangle_property_angle_sum(1,ABC)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
isosceles_triangle_judgment_line_equal(1,ADE)
isosceles_triangle_property_angle_equal(1,ADE)
triangle_property_angle_sum(1,ADE)
adjacent_complementary_angle(1,CED,DEA)
triangle_property_angle_sum(1,EDC)
Problem: As shown in the diagram, AB=5, AC=13, ∠ABC is the right angle of the right triangle ABC. Find the sum of the length of line AB and the length of line BC.
问题：如图所示，AB=5，AC=13，AB和BC是直角三角形ABC的两个直角边。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(AB),5)
Equal(LengthOfLine(AC),13)
RightTriangle(ABC)
Goal: Value(Add(LengthOfLine(AB),LengthOfLine(BC)))
Answer: 17
Theorem sequence:
right_triangle_property_pythagorean(1,ABC)
Problem: As shown in the diagram, ∠FGO=100°, quadrilateral OEFG is mirror congruent to quadrilateral OGCB. Find the measure of ∠FGH.
问题：如图所示，∠FGO=100°，OEFG镜像全等于OGCB。求∠FGH的大小。
Construction (CDL):
Shape(AD,DH,HE,EO,OA)
Shape(EH,HG,GO,OE)
Shape(HF,FG,GH)
Shape(OG,GC,CB,BO)
Collinear(AOB)
Collinear(DHGC)
Collinear(EHF)
Conditions (CDL):
Equal(MeasureOfAngle(FGO),100)
MirrorCongruentBetweenQuadrilateral(OEFG,OGCB)
Goal: Value(MeasureOfAngle(FGH))
Answer: 20
Theorem sequence:
mirror_congruent_quadrilateral_property_angle_equal(1,GOEF,GCBO)
adjacent_complementary_angle(1,HGO,OGC)
angle_addition(1,FGH,HGO)
Problem: As shown in the diagram, ∠DEC=∠BEA, ∠ECD=30°, △CDE is a right △. Find the measure of ∠BEA.
问题：如图所示，∠DEC=∠BEA，∠ECD=30°，△CDE为直角△。求∠BEA的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BC,CE)
Shape(EC,CD,DE)
Collinear(AED)
Conditions (CDL):
Equal(MeasureOfAngle(DEC),MeasureOfAngle(BEA))
Equal(MeasureOfAngle(ECD),30)
RightTriangle(CDE)
Goal: Value(MeasureOfAngle(BEA))
Answer: 60
Theorem sequence:
triangle_property_angle_sum(1,CDE)
Problem: As shown in the diagram, ∠CDB=40°, the center of circle O is O, PA is the tangent to ⊙O. Find the measure of ∠COB.
问题：如图所示，∠CDB=40°，O是圆O的圆心，圆O的切线为PA。求∠COB的大小。
Construction (CDL):
Shape(OBC,CE,EB)
Shape(OE,EC,CO)
Shape(BE,ED,DB)
Shape(ODB,BD)
Shape(OC,OCA,AO)
Shape(EO,OA,OAD,DE)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),40)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,D)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠ABC=30°, ∠CBF=20°, DE is parallel to BF, triangle CAB is a right triangle. Find the measure of ∠BAD.
问题：如图所示，∠ABC=30°，∠CBF=20°，DE∥BF，△CAB为直角△。求∠BAD的大小。
Construction (CDL):
Shape(DB,BA,AD)
Shape(AB,BC,CA)
Shape(AC,CB,BF,FE,EA)
Collinear(DAE)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),30)
Equal(MeasureOfAngle(CBF),20)
ParallelBetweenLine(DE,BF)
RightTriangle(CAB)
Goal: Value(MeasureOfAngle(BAD))
Answer: 50
Theorem sequence:
angle_addition(1,ABC,CBF)
parallel_property_collinear_extend(3,DE,BF,A)
parallel_property_ipsilateral_internal_angle(1,AE,BF)
adjacent_complementary_angle(1,EAB,BAD)
Problem: As shown in the diagram, ∠ABC=40°, the center of ⊙O is O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=40°，O是圆O的圆心。求∠AOC的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAB,BA)
Shape(DC,CA,AD)
Shape(DA,AB,BD)
Shape(CD,DO,OC)
Shape(OD,DB,OBC,CO)
Collinear(BDC)
Collinear(ADO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠BOA=52°, O is the center of circle O. Find the measure of ∠BDA.
问题：如图所示，∠BOA=52°，O是⊙O的圆心。求∠BDA的大小。
Construction (CDL):
Shape(DC,CO,OB,OBD)
Shape(OC,CB,BO)
Shape(CD,DA,AC)
Shape(BC,CA,OAB)
Shape(ODA,AD)
Collinear(DCB)
Collinear(ACO)
Cocircular(O,DAB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),52)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BDA))
Answer: 26
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,D)
Problem: As shown in the diagram, ∠FEA=40°, ∠GEF=∠DEG, ABCD is a rectangle. Find the measure of ∠FGE.
问题：如图所示，∠FEA=40°，∠GEF=∠DEG，四边形ABCD是长方形。求∠FGE的大小。
Construction (CDL):
Shape(AB,BF,FE,EA)
Shape(EF,FG,GE)
Shape(EG,GC,CD,DE)
Collinear(AED)
Collinear(BFGC)
Conditions (CDL):
Equal(MeasureOfAngle(FEA),40)
Equal(MeasureOfAngle(GEF),MeasureOfAngle(DEG))
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(FGE))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,DEF,FEA)
angle_addition(1,DEG,GEF)
angle_addition(1,GEF,FEA)
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_collinear_extend(3,CB,EA,G)
parallel_property_ipsilateral_internal_angle(1,GB,EA)
Problem: As shown in the diagram, ∠ECD=43°, CD is parallel to AB, BC is perpendicular to AC. Find the measure of ∠ABC.
问题：如图所示，∠ECD=43°，CD∥AB，BC垂直于AC。求∠ABC的大小。
Construction (CDL):
Shape(EC,CD)
Shape(DC,CB)
Shape(CA,AB,BC)
Collinear(ACE)
Conditions (CDL):
Equal(MeasureOfAngle(ECD),43)
ParallelBetweenLine(CD,AB)
PerpendicularBetweenLine(BC,AC)
Goal: Value(MeasureOfAngle(ABC))
Answer: 47
Theorem sequence:
adjacent_complementary_angle(1,ECB,BCA)
angle_addition(1,ECD,DCB)
parallel_property_alternate_interior_angle(1,CD,AB)
Problem: As shown in the diagram, AB=9, BC=6, D is the midpoint of segment CB, MA and MD are one pair of adjacent sides of the kite MAND. Find the perimeter of △DNB.
问题：如图所示，AB=9，BC=6，D是线段CB的中点，MA和MD是风筝形MAND的一组临边。求三角形DNB的周长。
Construction (CDL):
Shape(CM,MD,DC)
Shape(MA,AN,ND,DM)
Shape(DN,NB,BD)
Collinear(CMA)
Collinear(ANB)
Collinear(CDB)
Conditions (CDL):
Equal(LengthOfLine(AB),9)
Equal(LengthOfLine(BC),6)
IsMidpointOfLine(D,CB)
Kite(MAND)
Goal: Value(PerimeterOfTriangle(DNB))
Answer: 12
Theorem sequence:
line_addition(1,CD,DB)
line_addition(1,AN,NB)
triangle_perimeter_formula(1,DNB)
Problem: As shown in the diagram, ∠AOB=80°, OD bisects ∠AOC, OE bisects ∠COB. Find the measure of ∠DOE.
问题：如图所示，∠AOB=80°，OD是∠AOC的角平分线，OE是∠COB的角平分线。求∠DOE的大小。
Construction (CDL):
Shape(AO,OD)
Shape(DO,OC)
Shape(CO,OE)
Shape(EO,OB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),80)
IsBisectorOfAngle(OD,AOC)
IsBisectorOfAngle(OE,COB)
Goal: Value(MeasureOfAngle(DOE))
Answer: 40
Theorem sequence:
angle_addition(1,DOC,COE)
angle_addition(1,AOD,DOE)
angle_addition(1,AOE,EOB)
Problem: As shown in the diagram, BD=12, the perimeter of quadrilateral ABCD is 36, E bisects segment DC, AD and BC are opposite sides of the parallelogram ABCD. Find the perimeter of triangle DOE.
问题：如图所示，BD=12，ABCD的周长为36，E是线段DC的中点，BA和CD是▱ABCD的一组对边。求△DOE的周长。
Construction (CDL):
Shape(AB,BO,OA)
Shape(DA,AO,OD)
Shape(OB,BC,CO)
Shape(DO,OE,ED)
Shape(EO,OC,CE)
Collinear(AOC)
Collinear(BOD)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(BD),12)
Equal(PerimeterOfQuadrilateral(ABCD),36)
IsMidpointOfLine(E,DC)
Parallelogram(ABCD)
Goal: Value(PerimeterOfTriangle(DOE))
Answer: 15
Theorem sequence:
parallelogram_property_diagonal_bisection(1,BCDA,O)
line_addition(1,BO,OD)
midsegment_of_triangle_judgment_midpoint(1,OE,DBC)
midsegment_of_triangle_property_length(1,OE,DBC)
line_addition(1,CE,ED)
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
quadrilateral_perimeter_formula(1,ABCD)
triangle_perimeter_formula(1,DOE)
Problem: As shown in the diagram, AB=7, AC=3, AE is the angle bisector of ∠BAC, AC⊥BC, BD⊥ED. Find the length of line DB.
问题：如图所示，AB=7，AC=3，AE是∠BAC的角平分线，AC垂直于BC，BD垂直于ED。求直线DB的长度。
Construction (CDL):
Shape(AC,CE,EA)
Shape(AE,ED,DA)
Shape(DE,EB,BD)
Collinear(CEB)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),7)
Equal(LengthOfLine(AC),3)
IsBisectorOfAngle(AE,BAC)
PerpendicularBetweenLine(AC,BC)
PerpendicularBetweenLine(BD,ED)
Goal: Value(LengthOfLine(DB))
Answer: 4
Theorem sequence:
adjacent_complementary_angle(1,BDE,EDA)
bisector_of_angle_judgment_angle_equal(1,AE,DAC)
bisector_of_angle_property_distance_equal(1,AE,DAC)
mirror_congruent_triangle_judgment_hl(1,ACE,AED)
mirror_congruent_triangle_property_line_equal(1,EAC,EDA)
line_addition(1,AD,DB)
Problem: As shown in the diagram, ∠CAB=30°, ∠HBC=44°, DEFG is a rectangle, BC and CA are the legs of the right △ BCA. Find the measure of ∠BAF.
问题：如图所示，∠CAB=30°，∠HBC=44°，DEFG是长方形，BC和CA是直角三角形BCA的两个直角边。求∠BAF的大小。
Construction (CDL):
Shape(DE,EA,AH,HD)
Shape(CH,HB,BC)
Shape(HA,AB,BH)
Shape(BA,AF,FG,GB)
Shape(DH,HC)
Shape(CB,BG)
Collinear(DHBG)
Collinear(EAF)
Collinear(CHA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),30)
Equal(MeasureOfAngle(HBC),44)
Rectangle(DEFG)
RightTriangle(BCA)
Goal: Value(MeasureOfAngle(BAF))
Answer: 16
Theorem sequence:
triangle_property_angle_sum(1,ABC)
angle_addition(1,ABH,HBC)
parallel_property_collinear_extend(3,DG,EF,B)
parallel_property_collinear_extend(3,FE,BD,A)
parallel_property_alternate_interior_angle(2,DB,AF)
Problem: As shown in the diagram, ∠BCD=110°, the center of circle O is O. Find the measure of ∠DOB.
问题：如图所示，∠BCD=110°，O是⊙O的圆心。求∠DOB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(OD,DA,AB,BO)
Shape(OB,BC,CD,DO)
Shape(OBC,CB)
Shape(OCD,DC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_internal(1,OBD,C)
arc_property_center_angle(1,OBD,O)
Problem: As shown in the diagram, Sub(MeasureOfAngle(DBA)=MeasureOfAngle(CBE)), ∠ABC is the right angle of the right triangle ABC. Find the measure of ∠DBA.
问题：如图所示，Sub(MeasureOfAngle(DBA)=MeasureOfAngle(CBE))，∠ABC是直角三角形ABC的直角。求∠DBA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DB,BA)
Shape(CB,BE)
Collinear(DBE)
Conditions (CDL):
Equal(Sub(MeasureOfAngle(DBA),MeasureOfAngle(CBE)),54)
RightTriangle(ABC)
Goal: Value(MeasureOfAngle(DBA))
Answer: 72
Theorem sequence:
angle_addition(1,DBA,ABC)
adjacent_complementary_angle(1,DBC,CBE)
Problem: As shown in the diagram, △ABC is congruent to △DEC, ∠CAB=35°, ∠DCA=a°, ∠ECB=a°, ∠BCA is the right angle of the right triangle BCA. Find the value of a.
问题：如图所示，三角形ABC全等于三角形DEC，∠CAB=35°，∠DCA=a°，∠ECB=a°，三角形BCA为直角三角形。求a的值。
Construction (CDL):
Shape(DB,BF,FD)
Shape(FB,BC,CF)
Shape(AF,FC,CA)
Shape(BE,EC,CB)
Collinear(DFC)
Collinear(BFA)
Collinear(DBE)
Conditions (CDL):
CongruentBetweenTriangle(ABC,DEC)
Equal(MeasureOfAngle(CAB),35)
Equal(MeasureOfAngle(DCA),a)
Equal(MeasureOfAngle(ECB),a)
RightTriangle(BCA)
Goal: Value(a)
Answer: 70
Theorem sequence:
triangle_property_angle_sum(1,ABC)
congruent_triangle_property_line_equal(1,ABC,DEC)
congruent_triangle_property_angle_equal(1,BCA,ECD)
isosceles_triangle_judgment_line_equal(1,CBE)
isosceles_triangle_property_angle_equal(1,CBE)
triangle_property_angle_sum(1,CBE)