SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠DFE=135°, AB is parallel to CD. Find the measure of ∠FBA.
问题：如图所示，∠DFE=135°，AB∥CD。求∠FBA的大小。
Construction (CDL):
Shape(EF,FC)
Shape(DF,FE)
Shape(CF,FB)
Shape(BF,FD)
Shape(FB,BA)
Collinear(CFD)
Collinear(EFB)
Conditions (CDL):
Equal(MeasureOfAngle(DFE),135)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(FBA))
Answer: 45
Theorem sequence:
parallel_property_collinear_extend(3,DC,BA,F)
parallel_property_ipsilateral_internal_angle(1,FC,BA)
vertical_angle(1,CFB,DFE)
Problem: As shown in the diagram, ∠CBO=50°, the center of circle O is O. Find the measure of ∠BAC.
问题：如图所示，∠CBO=50°，O是⊙O的圆心。求∠BAC的大小。
Construction (CDL):
Shape(CO,OD,DA,OAC)
Shape(CD,DO,OC)
Shape(DC,CB,BD)
Shape(DB,BA,AD)
Shape(BC,OCB)
Shape(AB,OBA)
Collinear(ODB)
Collinear(CDA)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(CBO),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 40
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OCB)
isosceles_triangle_property_angle_equal(1,OCB)
triangle_property_angle_sum(1,OCB)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠ABD=20°, ∠BAE=45°, CA∥BD. Find the measure of ∠CAE.
问题：如图所示，∠ABD=20°，∠BAE=45°，CA平行于BD。求∠CAE的大小。
Construction (CDL):
Shape(AB,BD)
Shape(BA,AC)
Shape(CA,AE)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),20)
Equal(MeasureOfAngle(BAE),45)
ParallelBetweenLine(CA,BD)
Goal: Value(MeasureOfAngle(CAE))
Answer: 25
Theorem sequence:
parallel_property_alternate_interior_angle(2,CA,BD)
angle_addition(1,BAC,CAE)
Problem: As shown in the diagram, ∠ABC=40°, ∠CAB=90°, BC∥ED. Find the measure of ∠FDE.
问题：如图所示，∠ABC=40°，∠CAB=90°，BC平行于ED。求∠FDE的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BE,ED,DC,CB)
Shape(EG,GF,FD,DE)
Collinear(ABEG)
Collinear(ACDF)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),40)
Equal(MeasureOfAngle(CAB),90)
ParallelBetweenLine(BC,ED)
Goal: Value(MeasureOfAngle(FDE))
Answer: 130
Theorem sequence:
triangle_property_angle_sum(1,ABC)
parallel_property_corresponding_angle(2,DE,CB,A)
adjacent_complementary_angle(1,FDE,EDC)
Problem: As shown in the diagram, AB=7, AC=4, ∠ACD=90°, AD bisects ∠EAC, DE⊥AE. Find the length of line EB.
问题：如图所示，AB=7，AC=4，∠ACD=90°，AD是∠EAC的角平分线，DE垂直于AE。求直线EB的长度。
Construction (CDL):
Shape(BE,ED,DB)
Shape(DE,EA,AD)
Shape(DA,AC,CD)
Collinear(BDC)
Collinear(BEA)
Conditions (CDL):
Equal(LengthOfLine(AB),7)
Equal(LengthOfLine(AC),4)
Equal(MeasureOfAngle(ACD),90)
IsBisectorOfAngle(AD,EAC)
PerpendicularBetweenLine(DE,AE)
Goal: Value(LengthOfLine(EB))
Answer: 3
Theorem sequence:
mirror_congruent_triangle_judgment_aas(1,DAC,DEA)
mirror_congruent_triangle_property_line_equal(1,DAC,DEA)
line_addition(1,AE,EB)
Problem: As shown in the diagram, ∠COA=40°, O is the center of circle O. Find the measure of ∠CBA.
问题：如图所示，∠COA=40°，O是⊙O的圆心。求∠CBA的大小。
Construction (CDL):
Shape(AO,OD,DB,OBA)
Shape(AD,DO,OA)
Shape(DC,CB,BD)
Shape(CD,DA,OAC)
Shape(BC,OCB)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 20
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, ∠OBA=30°, O is the center of circle O. Find the measure of ∠ACB.
问题：如图所示，∠OBA=30°，O是⊙O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(CD,DO,OA,OAC)
Shape(CB,BD,DC)
Shape(OD,DA,AO)
Shape(AD,DB,BA)
Shape(BC,OCB)
Shape(AB,OBA)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(OBA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 60
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,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,OBA)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠CAB=50°, O is the center of ⊙O, BA is the diameter of circle O. Find the measure of ∠ADC.
问题：如图所示，∠CAB=50°，⊙O的圆心为O，⊙O的直径为BA。求∠ADC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AE,EC,CA)
Shape(CE,EO,OB,BC)
Shape(CB,OBC)
Shape(DA,OAD)
Shape(AD,DE,EA)
Shape(BO,OE,ED,ODB)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,BCAD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BA,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠AOB=70°, ∠ECA=25°, △ODA is mirror congruent to △OBC. Find the measure of ∠DAO.
问题：如图所示，∠AOB=70°，∠ECA=25°，三角形ODA镜像全等于三角形OBC。求∠DAO的大小。
Construction (CDL):
Shape(OB,BE,EA,AO)
Shape(EB,BD,DE)
Shape(AE,EC,CA)
Collinear(OBD)
Collinear(AED)
Collinear(OAC)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),70)
Equal(MeasureOfAngle(ECA),25)
MirrorCongruentBetweenTriangle(ODA,OBC)
Goal: Value(MeasureOfAngle(DAO))
Answer: 85
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,DAO,COB)
triangle_property_angle_sum(1,ODA)
Problem: As shown in the diagram, ∠CBA=40°, the center of ⊙O is O. Find the measure of ∠COA.
问题：如图所示，∠CBA=40°，O是⊙O的圆心。求∠COA的大小。
Construction (CDL):
Shape(AO,OD,DB,OBA)
Shape(AD,DO,OA)
Shape(DC,CB,BD)
Shape(CD,DA,AC)
Shape(CA,OAC)
Shape(BC,OCB)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, AB=36, AC=24, BO is the angle bisector of ∠MBC, CO bisects ∠BCN, MN is parallel to BC. Find the perimeter of triangle AMN.
问题：如图所示，AB=36，AC=24，BO是∠MBC的角平分线，CO是∠BCN的角平分线，MN平行于BC。求三角形AMN的周长。
Construction (CDL):
Shape(AM,MO,ON,NA)
Shape(OM,MB,BO)
Shape(OB,BC,CO)
Shape(OC,CN,NO)
Collinear(AMB)
Collinear(MON)
Collinear(ANC)
Conditions (CDL):
Equal(LengthOfLine(AB),36)
Equal(LengthOfLine(AC),24)
IsBisectorOfAngle(BO,MBC)
IsBisectorOfAngle(CO,BCN)
ParallelBetweenLine(MN,BC)
Goal: Value(PerimeterOfTriangle(AMN))
Answer: 60
Theorem sequence:
parallel_property_collinear_extend(3,MN,BC,O)
parallel_property_alternate_interior_angle(2,MO,BC)
parallel_property_alternate_interior_angle(1,ON,BC)
isosceles_triangle_judgment_angle_equal(1,MBO)
isosceles_triangle_judgment_angle_equal(1,NOC)
line_addition(1,AM,MB)
line_addition(1,AN,NC)
line_addition(1,MO,ON)
triangle_perimeter_formula(1,AMN)
Problem: As shown in the diagram, ∠ACJ=130°, AB∥EF. Find the measure of ∠FDK.
问题：如图所示，∠ACJ=130°，AB平行于EF。求∠FDK的大小。
Construction (CDL):
Shape(AC,CJ)
Shape(JC,CB)
Shape(DC,CA)
Shape(BC,CD)
Shape(ED,DC)
Shape(CD,DF)
Shape(KD,DE)
Shape(FD,DK)
Collinear(ACB)
Collinear(EDF)
Collinear(JCDK)
Conditions (CDL):
Equal(MeasureOfAngle(ACJ),130)
ParallelBetweenLine(AB,EF)
Goal: Value(MeasureOfAngle(FDK))
Answer: 130
Theorem sequence:
parallel_property_collinear_extend(3,AB,EF,C)
parallel_property_collinear_extend(3,FE,CA,D)
parallel_property_corresponding_angle(2,DE,CA,J)
vertical_angle(1,EDC,FDK)
Problem: As shown in the diagram, ∠ACB=90°, ∠EDA=46°, AB is parallel to ED. Find the measure of ∠CBA.
问题：如图所示，∠ACB=90°，∠EDA=46°，AB∥ED。求∠CBA的大小。
Construction (CDL):
Shape(ED,DA)
Shape(CD,DE)
Shape(AD,DC,CB,BA)
Collinear(ADC)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(EDA),46)
ParallelBetweenLine(AB,ED)
Goal: Value(MeasureOfAngle(CBA))
Answer: 44
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,ED)
triangle_property_angle_sum(1,ACB)
Problem: As shown in the diagram, BD=120, CD=60, EC=50, AB⊥DB, EC is perpendicular to DC. Find the length of line AB.
问题：如图所示，BD=120，CD=60，EC=50，AB⊥DB，EC⊥DC。求直线AB的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DE,EC,CD)
Collinear(ADE)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(BD),120)
Equal(LengthOfLine(CD),60)
Equal(LengthOfLine(EC),50)
PerpendicularBetweenLine(AB,DB)
PerpendicularBetweenLine(EC,DC)
Goal: Value(LengthOfLine(AB))
Answer: 100
Theorem sequence:
vertical_angle(1,CDE,BDA)
similar_triangle_judgment_aa(1,ABD,ECD)
similar_triangle_property_line_ratio(1,ABD,ECD)
similar_triangle_property_line_ratio(1,DAB,DEC)
Problem: As shown in the diagram, CD=CE, ∠CDE=74°, DC∥BA. Find the measure of ∠CBA.
问题：如图所示，CD=CE，∠CDE=74°，DC∥BA。求∠CBA的大小。
Construction (CDL):
Shape(DE,EC,CD)
Shape(CE,EB,EA)
Collinear(CEB)
Conditions (CDL):
Equal(LengthOfLine(CD),LengthOfLine(CE))
Equal(MeasureOfAngle(CDE),74)
ParallelBetweenLine(DC,BA)
Goal: Value(MeasureOfAngle(CBA))
Answer: 32
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,CDE)
isosceles_triangle_property_angle_equal(1,CDE)
triangle_property_angle_sum(1,CDE)
parallel_property_alternate_interior_angle(2,DC,BA)
Problem: As shown in the diagram, ∠ADE=145°, ∠CAB=90°, CB∥ED. Find the measure of ∠ABC.
问题：如图所示，∠ADE=145°，∠CAB=90°，CB平行于ED。求∠ABC的大小。
Construction (CDL):
Shape(ED,DC)
Shape(AD,DE)
Shape(CD,DA,AB,BC)
Collinear(ADC)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),145)
Equal(MeasureOfAngle(CAB),90)
ParallelBetweenLine(CB,ED)
Goal: Value(MeasureOfAngle(ABC))
Answer: 55
Theorem sequence:
parallel_property_alternate_interior_angle(1,CB,ED)
triangle_property_angle_sum(1,ABC)
adjacent_complementary_angle(1,ADE,EDC)
Problem: As shown in the diagram, ∠CBA=40°, AB∥CD, CB is parallel to DE. Find the measure of ∠CDE.
问题：如图所示，∠CBA=40°，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),40)
ParallelBetweenLine(AB,CD)
ParallelBetweenLine(CB,DE)
Goal: Value(MeasureOfAngle(CDE))
Answer: 140
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CD)
parallel_property_ipsilateral_internal_angle(1,CB,DE)
Problem: As shown in the diagram, ∠ACG=120°, ∠BAD=80°, DE is parallel to FG. Find the measure of ∠CAB.
问题：如图所示，∠ACG=120°，∠BAD=80°，DE平行于FG。求∠CAB的大小。
Construction (CDL):
Shape(BA,AD)
Shape(FB,BA)
Shape(EA,AC)
Shape(AC,CG)
Shape(AB,BC,CA)
Collinear(DAE)
Collinear(FBCG)
Conditions (CDL):
Equal(MeasureOfAngle(ACG),120)
Equal(MeasureOfAngle(BAD),80)
ParallelBetweenLine(DE,FG)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
parallel_property_collinear_extend(3,DE,FG,A)
parallel_property_collinear_extend(3,GF,EA,C)
parallel_property_ipsilateral_internal_angle(1,AE,CG)
angle_addition(1,CAB,BAD)
adjacent_complementary_angle(1,EAC,CAD)
Problem: As shown in the diagram, ∠EAB=40°, AB∥DC. Find the measure of ∠DCA.
问题：如图所示，∠EAB=40°，AB平行于DC。求∠DCA的大小。
Construction (CDL):
Shape(DC,CA)
Shape(CA,AF)
Shape(BA,AC)
Shape(EA,AB)
Shape(FA,AE)
Collinear(CAE)
Collinear(FAB)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),40)
ParallelBetweenLine(AB,DC)
Goal: Value(MeasureOfAngle(DCA))
Answer: 140
Theorem sequence:
adjacent_complementary_angle(1,FAE,EAB)
parallel_property_collinear_extend(1,AB,DC,F)
parallel_property_corresponding_angle(2,CD,AF,E)
Problem: As shown in the diagram, ∠OAB=34°, OF is the angle bisector of ∠EOD, OD∥AB. Find the measure of ∠EOD.
问题：如图所示，∠OAB=34°，OF平分∠EOD，OD平行于AB。求∠EOD的大小。
Construction (CDL):
Shape(AO,OC)
Shape(CO,OE)
Shape(EO,OF)
Shape(OA,AB)
Shape(DO,OA)
Shape(FO,OD)
Collinear(AOF)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),34)
IsBisectorOfAngle(OF,EOD)
ParallelBetweenLine(OD,AB)
Goal: Value(MeasureOfAngle(EOD))
Answer: 68
Theorem sequence:
parallel_property_corresponding_angle(1,OD,AB,F)
angle_addition(1,EOF,FOD)
Problem: As shown in the diagram, AB=80, BC=60, EF=900, AB⊥CB, DE is perpendicular to FE, triangle ABC is similar to triangle DEF.. Find the length of line DE.
问题：如图所示，AB=80，BC=60，EF=900，AB垂直于CB，DE⊥FE，三角形ABC相似于三角形DEF。求直线DE的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(LengthOfLine(AB),80)
Equal(LengthOfLine(BC),60)
Equal(LengthOfLine(EF),900)
PerpendicularBetweenLine(AB,CB)
PerpendicularBetweenLine(DE,FE)
SimilarBetweenTriangle(ABC,DEF)
Goal: Value(LengthOfLine(DE))
Answer: 1200
Theorem sequence:
similar_triangle_property_line_ratio(1,ABC,DEF)
similar_triangle_property_line_ratio(1,CAB,FDE)
Problem: As shown in the diagram, ∠DCB=110°. Find the measure of ∠BAD.
问题：如图所示，∠DCB=110°。求∠BAD的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(DA,OAD)
Shape(CD,ODC)
Shape(BC,OCB)
Shape(CB,BD,DC)
Shape(BA,AD,DB)
Cocircular(O,CBAD)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),110)
Goal: Value(MeasureOfAngle(BAD))
Answer: 70
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,C)
arc_property_circumference_angle_internal(1,OBD,A)
Problem: As shown in the diagram, ∠BED=40°, EA bisects ∠CEB, AB is parallel to CD. Find the measure of ∠BAE.
问题：如图所示，∠BED=40°，EA平分∠CEB，AB平行于CD。求∠BAE的大小。
Construction (CDL):
Shape(CE,EA)
Shape(AE,EB,BA)
Shape(BE,ED)
Collinear(CED)
Conditions (CDL):
Equal(MeasureOfAngle(BED),40)
IsBisectorOfAngle(EA,CEB)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BAE))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,CEB,BED)
angle_addition(1,CEA,AEB)
parallel_property_collinear_extend(3,DC,BA,E)
parallel_property_alternate_interior_angle(1,AB,CE)
Problem: As shown in the diagram, ∠ABC=90°, ∠ACF=23°, ∠CAB=60°, quadrilateral DEFG is a rectangle. Find the measure of ∠HAB.
问题：如图所示，∠ABC=90°，∠ACF=23°，∠CAB=60°，四边形DEFG是矩形。求∠HAB的大小。
Construction (CDL):
Shape(DE,EC,CH,HD)
Shape(HC,CA,AH)
Shape(AC,CF,FG,GA)
Shape(BH,HA,AB)
Collinear(DHAG)
Collinear(ECF)
Collinear(BHC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),90)
Equal(MeasureOfAngle(ACF),23)
Equal(MeasureOfAngle(CAB),60)
Rectangle(DEFG)
Goal: Value(MeasureOfAngle(HAB))
Answer: 37
Theorem sequence:
parallel_property_collinear_extend(3,DG,EF,A)
parallel_property_collinear_extend(3,FE,AD,C)
parallel_property_alternate_interior_angle(2,DA,CF)
angle_addition(1,CAH,HAB)
Problem: As shown in the diagram, AB=AC, ∠CAB=100°, AD∥BC. Find the measure of ∠DAC.
问题：如图所示，AB=AC，∠CAB=100°，AD平行于BC。求∠DAC的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DA,AC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),100)
ParallelBetweenLine(AD,BC)
Goal: Value(MeasureOfAngle(DAC))
Answer: 40
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
parallel_property_alternate_interior_angle(1,AD,BC)
Problem: As shown in the diagram, ∠ABE=70°, AB∥CD. Find the measure of ∠ECD.
问题：如图所示，∠ABE=70°，AB∥CD。求∠ECD的大小。
Construction (CDL):
Shape(AB,BE)
Shape(AC,CB,BA)
Shape(BC,CD)
Collinear(EBC)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),70)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(ECD))
Answer: 110
Theorem sequence:
adjacent_complementary_angle(1,CBA,ABE)
parallel_property_alternate_interior_angle(2,AB,CD)
Problem: As shown in the diagram, the diameter of ⊙O is 200, AB=160, O is the center of ⊙O, AC⊥OC. Find the length of line CD.
问题：如图所示，⊙O的直径为200，AB=160，O是圆O的圆心，AC垂直于OC。求直线CD的长度。
Construction (CDL):
Shape(OA,AC,CO)
Shape(OC,CB,BO)
Shape(AO,OB,OBA)
Shape(DC,CA,OAD)
Shape(BC,CD,ODB)
Collinear(OCD)
Collinear(ACB)
Cocircular(O,ADB)
Conditions (CDL):
Equal(DiameterOfCircle(O),200)
Equal(LengthOfLine(AB),160)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AC,OC)
Goal: Value(LengthOfLine(CD))
Answer: 40
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,O)
radius_of_circle_property_length_equal(1,OA,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OC,AB)
line_addition(1,AC,CB)
right_triangle_judgment_angle(1,ACO)
right_triangle_property_pythagorean(1,ACO)
radius_of_circle_property_length_equal(1,OD,O)
line_addition(1,OC,CD)
Problem: As shown in the diagram, ∠ABF=40°, ∠CAB is the right angle of the isosceles right △ BCA, DE is parallel to FB. Find the measure of ∠DEC.
问题：如图所示，∠ABF=40°，∠ABC是等腰直角三角形BCA的直角，DE平行于FB。求∠DEC的大小。
Construction (CDL):
Shape(CD,DE,EC)
Shape(DF,FB,BE,ED)
Shape(FA,AB,BF)
Collinear(CDFA)
Collinear(CEB)
Conditions (CDL):
Equal(MeasureOfAngle(ABF),40)
IsoscelesRightTriangle(BCA)
ParallelBetweenLine(DE,FB)
Goal: Value(MeasureOfAngle(DEC))
Answer: 50
Theorem sequence:
angle_addition(1,ABF,FBE)
parallel_property_corresponding_angle(2,BF,ED,C)
Problem: As shown in the diagram, ∠ABC=150°, ∠BAM=120°, BE is parallel to CN, MA∥BE. Find the measure of ∠BCN.
问题：如图所示，∠ABC=150°，∠BAM=120°，BE∥CN，MA平行于BE。求∠BCN的大小。
Construction (CDL):
Shape(BA,AM)
Shape(AB,BE)
Shape(EB,BC)
Shape(BC,CN)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),150)
Equal(MeasureOfAngle(BAM),120)
ParallelBetweenLine(BE,CN)
ParallelBetweenLine(MA,BE)
Goal: Value(MeasureOfAngle(BCN))
Answer: 150
Theorem sequence:
parallel_property_alternate_interior_angle(2,MA,BE)
angle_addition(1,ABE,EBC)
parallel_property_ipsilateral_internal_angle(1,BE,CN)
Problem: As shown in the diagram, EF=23, E bisects segment AB, F is the midpoint of segment BC. Find the length of line AC.
问题：如图所示，EF=23，E平分线段AB，F是线段BC的中点。求直线AC的长度。
Construction (CDL):
Shape(BF,FE,EB)
Shape(EF,FC,CA,AE)
Collinear(BEA)
Collinear(BFC)
Conditions (CDL):
Equal(LengthOfLine(EF),23)
IsMidpointOfLine(E,AB)
IsMidpointOfLine(F,BC)
Goal: Value(LengthOfLine(AC))
Answer: 46
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,FE,BCA)
midsegment_of_triangle_property_length(1,FE,BCA)
Problem: As shown in the diagram, AB=AD, AD=DC, ∠ABD=80°. Find the measure of ∠DCA.
问题：如图所示，AB=AD，AD=DC，∠ABD=80°。求∠DCA的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AD))
Equal(LengthOfLine(AD),LengthOfLine(DC))
Equal(MeasureOfAngle(ABD),80)
Goal: Value(MeasureOfAngle(DCA))
Answer: 40
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,ABD)
adjacent_complementary_angle(1,BDA,ADC)
isosceles_triangle_judgment_line_equal(1,DCA)
isosceles_triangle_property_angle_equal(1,DCA)
triangle_property_angle_sum(1,ADC)
Problem: As shown in the diagram, ∠DCB=70°, BD is the angle bisector of ∠CBA, CD∥BA. Find the measure of ∠BDC.
问题：如图所示，∠DCB=70°，BD平分∠CBA，CD平行于BA。求∠BDC的大小。
Construction (CDL):
Shape(CB,BD,DC)
Shape(DB,BA)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),70)
IsBisectorOfAngle(BD,CBA)
ParallelBetweenLine(CD,BA)
Goal: Value(MeasureOfAngle(BDC))
Answer: 55
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,CD,BA)
angle_addition(1,CBD,DBA)
parallel_property_alternate_interior_angle(2,CD,BA)
Problem: As shown in the diagram, ∠ADE=38°, AD bisects ∠EAB, CD is parallel to AB. Find the measure of ∠AEC.
问题：如图所示，∠ADE=38°，AD平分∠EAB，CD∥AB。求∠AEC的大小。
Construction (CDL):
Shape(AE,EC)
Shape(EA,AD,DE)
Shape(DA,AB)
Collinear(CED)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),38)
IsBisectorOfAngle(AD,EAB)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(AEC))
Answer: 76
Theorem sequence:
parallel_property_alternate_interior_angle(2,CD,AB)
triangle_property_angle_sum(1,ADE)
adjacent_complementary_angle(1,DEA,AEC)
Problem: As shown in the diagram, ∠DAB=36°, O is the center of ⊙O, AB is the tangent to circle O. Find the measure of ∠BCO.
问题：如图所示，∠DAB=36°，⊙O的圆心为O，圆O的切线为AB。求∠BCO的大小。
Construction (CDL):
Shape(DA,AB,ODB)
Shape(BO,OD,ODB)
Shape(OB,BC,CO)
Shape(CB,OBC)
Shape(DO,OC,OCD)
Collinear(ADOC)
Cocircular(O,DBC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),36)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 27
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AB,O,O)
triangle_property_angle_sum(1,ABO)
adjacent_complementary_angle(1,COB,BOA)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, ∠BCD=80°. Find the measure of ∠DAB.
问题：如图所示，∠BCD=80°。求∠DAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AD,ODA)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(AB,BD,DA)
Shape(BC,CD,DB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),80)
Goal: Value(MeasureOfAngle(DAB))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_internal(1,ODB,A)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠ABC=50°, ∠BCA=80°, BP is the angle bisector of ∠ABC, CP is the angle bisector of ∠BCA. Find the measure of ∠CPB.
问题：如图所示，∠ABC=50°，∠BCA=80°，BP平分∠ABC，CP是∠BCA的角平分线。求∠CPB的大小。
Construction (CDL):
Shape(AB,BP,PC,CA)
Shape(PB,BC,CP)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
Equal(MeasureOfAngle(BCA),80)
IsBisectorOfAngle(BP,ABC)
IsBisectorOfAngle(CP,BCA)
Goal: Value(MeasureOfAngle(CPB))
Answer: 115
Theorem sequence:
angle_addition(1,ABP,PBC)
angle_addition(1,BPC,PCA)
triangle_property_angle_sum(1,CPB)
Problem: As shown in the diagram, ∠CNG=65°, DG∥EF, AB is perpendicular to CB. Find the measure of ∠EBM.
问题：如图所示，∠CNG=65°，DG∥EF，AB⊥CB。求∠EBM的大小。
Construction (CDL):
Shape(CA,AM,MN,NC)
Shape(MD,DE,EB,BM)
Shape(MB,BN,NM)
Shape(NB,BF,FG,GN)
Collinear(DMNG)
Collinear(EBF)
Collinear(AMB)
Collinear(CNB)
Conditions (CDL):
Equal(MeasureOfAngle(CNG),65)
ParallelBetweenLine(DG,EF)
PerpendicularBetweenLine(AB,CB)
Goal: Value(MeasureOfAngle(EBM))
Answer: 25
Theorem sequence:
parallel_property_collinear_extend(3,DG,EF,N)
parallel_property_collinear_extend(3,FE,GN,B)
parallel_property_corresponding_angle(1,NG,BF,C)
angle_addition(1,MBN,NBF)
adjacent_complementary_angle(1,EBM,MBF)
Problem: As shown in the diagram, ∠GEB=56°, AB∥CD. Find the measure of ∠EFD.
问题：如图所示，∠GEB=56°，AB∥CD。求∠EFD的大小。
Construction (CDL):
Shape(AE,EG)
Shape(GE,EB)
Shape(FE,EA)
Shape(BE,EF)
Shape(CF,FE)
Shape(EF,FD)
Shape(HF,FC)
Shape(DF,FH)
Collinear(AEB)
Collinear(CFD)
Collinear(HFEG)
Conditions (CDL):
Equal(MeasureOfAngle(GEB),56)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(EFD))
Answer: 56
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,E)
parallel_property_collinear_extend(3,DC,BE,F)
parallel_property_corresponding_angle(1,EB,FD,G)
Problem: As shown in the diagram, BC=DiameterOfCircle(O), ∠EDC=36°, OE is perpendicular to AE. Find the measure of ∠DAB.
问题：如图所示，BC=DiameterOfCircle(O)，∠EDC=36°，OE⊥AE。求∠DAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OE,EA)
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(DE,EO,OB,OBD)
Shape(ED,DC,CE)
Shape(CD,ODC)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(LengthOfLine(BC),DiameterOfCircle(O))
Equal(MeasureOfAngle(EDC),36)
PerpendicularBetweenLine(OE,AE)
Goal: Value(MeasureOfAngle(DAB))
Answer: 54
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
triangle_property_angle_sum(1,ABE)
Problem: As shown in the diagram, ∠OPD=70°, AB∥CD. Find the measure of ∠BOF.
问题：如图所示，∠OPD=70°，AB∥CD。求∠BOF的大小。
Construction (CDL):
Shape(AO,OE)
Shape(EO,OB)
Shape(PO,OA)
Shape(BO,OP)
Shape(CP,PO)
Shape(OP,PD)
Shape(FP,PC)
Shape(DP,PF)
Collinear(AOB)
Collinear(CPD)
Collinear(EOPF)
Conditions (CDL):
Equal(MeasureOfAngle(OPD),70)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BOF))
Answer: 110
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,O)
parallel_property_collinear_extend(3,DC,BO,P)
parallel_property_ipsilateral_internal_angle(1,OB,PD)
Problem: As shown in the diagram, ∠BOA=50°. Find the measure of ∠DOC.
问题：如图所示，∠BOA=50°。求∠DOC的大小。
Construction (CDL):
Shape(BO,OA)
Shape(AO,OD)
Shape(CO,OB)
Shape(DO,OC)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),50)
Goal: Value(MeasureOfAngle(DOC))
Answer: 50
Theorem sequence:
vertical_angle(1,BOA,DOC)
Problem: As shown in the diagram, ∠CAB=35°, the center of ⊙O is O, BA is the diameter of circle O. Find the measure of ∠ABC.
问题：如图所示，∠CAB=35°，O是⊙O的圆心，⊙O的直径为BA。求∠ABC的大小。
Construction (CDL):
Shape(AO,OB,BC,CA)
Shape(BO,OA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BA,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 55
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)