SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=8, BC=6, ED is the perpendicular bisector of AC. Find the perimeter of triangle EBC.
问题：如图所示，AB=8，BC=6，ED是AC的垂直平分线。求三角形EBC的周长。
Construction (CDL):
Shape(AE,ED,DA)
Shape(DE,EC,CD)
Shape(EB,BC,CE)
Collinear(AEB)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(BC),6)
IsPerpendicularBisectorOfLine(ED,AC)
Goal: Value(PerimeterOfTriangle(EBC))
Answer: 14
Theorem sequence:
line_addition(1,AE,EB)
perpendicular_bisector_property_distance_equal(1,ED,AC)
triangle_perimeter_formula(1,EBC)
Problem: As shown in the diagram, ∠CBD=60°, AB∥FE. Find the measure of ∠FEB.
问题：如图所示，∠CBD=60°，AB平行于FE。求∠FEB的大小。
Construction (CDL):
Shape(AB,BC)
Shape(CB,BD)
Shape(DB,BE)
Shape(EB,BA)
Shape(FE,EB)
Shape(BE,EG)
Shape(GE,EH)
Shape(HE,EF)
Collinear(ABD)
Collinear(FEG)
Collinear(CBEH)
Conditions (CDL):
Equal(MeasureOfAngle(CBD),60)
ParallelBetweenLine(AB,FE)
Goal: Value(MeasureOfAngle(FEB))
Answer: 120
Theorem sequence:
vertical_angle(1,CBD,EBA)
parallel_property_ipsilateral_internal_angle(1,EF,BA)
Problem: As shown in the diagram, ∠DEF=∠FEG, ∠MFE=60°, ABCD is a rectangle. Find the measure of ∠GEA.
问题：如图所示，∠DEF=∠FEG，∠MFE=60°，四边形ABCD是矩形。求∠GEA的大小。
Construction (CDL):
Shape(AB,BM,MG,GE,EA)
Shape(DE,EF,FC,CD)
Shape(EG,GM,MF,FE)
Shape(FM,MH,HF)
Collinear(BMFC)
Collinear(AED)
Collinear(GMH)
Conditions (CDL):
Equal(MeasureOfAngle(DEF),MeasureOfAngle(FEG))
Equal(MeasureOfAngle(MFE),60)
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(GEA))
Answer: 60
Theorem sequence:
angle_addition(1,DEF,FEG)
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_collinear_extend(3,CB,DE,F)
parallel_property_alternate_interior_angle(1,FB,DE)
adjacent_complementary_angle(1,DEG,GEA)
Problem: As shown in the diagram, ∠BCF=∠FHG, ∠EBC=∠HGE, ∠FAE=60°, ∠GEA=85°. Find the measure of ∠AFH.
问题：如图所示，∠BCF=∠FHG，∠EBC=∠HGE，∠FAE=60°，∠GEA=85°。求∠AFH的大小。
Construction (CDL):
Shape(AE,EF,FA)
Shape(FE,EB,BC,CF)
Shape(EA,AF,FH,HG,GE)
Collinear(AEB)
Collinear(AFC)
Conditions (CDL):
Equal(MeasureOfAngle(BCF),MeasureOfAngle(FHG))
Equal(MeasureOfAngle(EBC),MeasureOfAngle(HGE))
Equal(MeasureOfAngle(FAE),60)
Equal(MeasureOfAngle(GEA),85)
Goal: Value(MeasureOfAngle(AFH))
Answer: 35
Theorem sequence:
angle_addition(1,GEA,AEF)
angle_addition(1,EFA,AFH)
triangle_property_angle_sum(1,AEF)
triangle_property_angle_sum(1,ABC)
quadrilateral_property_angle_sum(1,GEFH)
Problem: As shown in the diagram, Mul(LengthOfLine(BF)=LengthOfLine(BF)), Mul(LengthOfLine(CF)=LengthOfLine(CF)), FC and CB are the legs of the right △ FCB, ABCD is a square, ECFG is a square, FBHI is a square. Find the area of quadrilateral ABCD.
问题：如图所示，Mul(LengthOfLine(BF)=LengthOfLine(BF))，Mul(LengthOfLine(CF)=LengthOfLine(CF))，∠FCB是直角△FCB的直角，四边形ABCD是正方形，ECFG是正方形，FBHI是正方形。求ABCD的面积。
Construction (CDL):
Shape(AB,BC,CD,DA)
Shape(EC,CF,FG,GE)
Shape(CB,BF,FC)
Shape(FB,BH,HI,IF)
Collinear(DCF)
Collinear(BCG)
Conditions (CDL):
Equal(Mul(LengthOfLine(BF),LengthOfLine(BF)),169)
Equal(Mul(LengthOfLine(CF),LengthOfLine(CF)),144)
RightTriangle(FCB)
Square(ABCD)
Square(ECFG)
Square(FBHI)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 25
Theorem sequence:
right_triangle_property_pythagorean(1,FCB)
parallelogram_area_formula_sine(1,ABCD)
Problem: As shown in the diagram, ∠ABC=60°, DF is parallel to AE, CB is perpendicular to DB. Find the measure of ∠GDF.
问题：如图所示，∠ABC=60°，DF平行于AE，CB⊥DB。求∠GDF的大小。
Construction (CDL):
Shape(AB,BC)
Shape(CB,BD,DC)
Shape(DB,BE)
Shape(GD,DF)
Shape(FD,DB)
Shape(CD,DG)
Collinear(ABE)
Collinear(BDG)
Collinear(CDF)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),60)
ParallelBetweenLine(DF,AE)
PerpendicularBetweenLine(CB,DB)
Goal: Value(MeasureOfAngle(GDF))
Answer: 30
Theorem sequence:
angle_addition(1,ABC,CBD)
adjacent_complementary_angle(1,ABD,DBE)
parallel_property_collinear_extend(3,EA,FD,B)
parallel_property_corresponding_angle(1,DF,BE,G)
Problem: As shown in the diagram, ∠ABC=∠BCA, ∠CAB=50°, ∠PCA=∠PBC. Find the measure of ∠CPB.
问题：如图所示，∠ABC=∠BCA，∠CAB=50°，∠PCA=∠PBC。求∠CPB的大小。
Construction (CDL):
Shape(AB,BP,PC,CA)
Shape(CP,PB,BC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),MeasureOfAngle(BCA))
Equal(MeasureOfAngle(CAB),50)
Equal(MeasureOfAngle(PCA),MeasureOfAngle(PBC))
Goal: Value(MeasureOfAngle(CPB))
Answer: 115
Theorem sequence:
angle_addition(1,BCP,PCA)
triangle_property_angle_sum(1,CPB)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠BEF=56°, ∠BEF=∠FEG, ABCD is a rectangle. Find the measure of ∠EGF.
问题：如图所示，∠BEF=56°，∠BEF=∠FEG，ABCD是矩形。求∠EGF的大小。
Construction (CDL):
Shape(AB,BE,EF,FA)
Shape(GE,EC,CD,DG)
Shape(FE,EG,GF)
Collinear(AFGD)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(BEF),56)
Equal(MeasureOfAngle(BEF),MeasureOfAngle(FEG))
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(EGF))
Answer: 68
Theorem sequence:
angle_addition(1,BEF,FEG)
parallel_property_collinear_extend(3,AD,BC,G)
parallel_property_collinear_extend(3,CB,GA,E)
parallel_property_ipsilateral_internal_angle(1,EB,GA)
Problem: As shown in the diagram, ∠AOE=24°. Find the measure of ∠EOB.
问题：如图所示，∠AOE=24°。求∠EOB的大小。
Construction (CDL):
Shape(AO,OE)
Shape(EO,OD)
Shape(DO,OB)
Shape(BO,OF)
Shape(FO,OC)
Shape(CO,OA)
Collinear(AOB)
Collinear(COD)
Collinear(EOF)
Conditions (CDL):
Equal(MeasureOfAngle(AOE),24)
Goal: Value(MeasureOfAngle(EOB))
Answer: 156
Theorem sequence:
adjacent_complementary_angle(1,AOE,EOB)
Problem: As shown in the diagram, ∠DCA=20°, O is the center of ⊙O. Find the measure of ∠BOA.
问题：如图所示，∠DCA=20°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(CA,AD,DC)
Shape(OD,DB,BO)
Shape(AC,OCA)
Shape(BD,DA,OAB)
Shape(CD,DO,OB,OBC)
Collinear(CDB)
Collinear(ADO)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(DCA),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠CBD=72°, AB is parallel to FE. Find the measure of ∠FEB.
问题：如图所示，∠CBD=72°，AB∥FE。求∠FEB的大小。
Construction (CDL):
Shape(AB,BC)
Shape(CB,BD)
Shape(DB,BE)
Shape(EB,BA)
Shape(FE,EB)
Shape(BE,EG)
Shape(GE,EH)
Shape(HE,EF)
Collinear(ABD)
Collinear(FEG)
Collinear(CBEH)
Conditions (CDL):
Equal(MeasureOfAngle(CBD),72)
ParallelBetweenLine(AB,FE)
Goal: Value(MeasureOfAngle(FEB))
Answer: 108
Theorem sequence:
vertical_angle(1,CBD,EBA)
parallel_property_ipsilateral_internal_angle(1,EF,BA)
Problem: As shown in the diagram, ∠ABC=120°, ∠GCE=∠ECB, quadrilateral AFGD is a rectangle. Find the measure of ∠ECB.
问题：如图所示，∠ABC=120°，∠GCE=∠ECB，四边形AFGD是长方形。求∠ECB的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Shape(CB,BE,EC)
Shape(CE,EF,FG,GC)
Collinear(ABEF)
Collinear(DCG)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),120)
Equal(MeasureOfAngle(GCE),MeasureOfAngle(ECB))
Rectangle(AFGD)
Goal: Value(MeasureOfAngle(ECB))
Answer: 60
Theorem sequence:
parallel_property_collinear_extend(3,FA,GD,B)
parallel_property_collinear_extend(3,DG,AB,C)
parallel_property_alternate_interior_angle(1,CG,AB)
angle_addition(1,GCE,ECB)
Problem: As shown in the diagram, ∠BPA=126°, ∠MQF=100°. Find the measure of ∠MAP minus the measure of ∠QFM.
问题：如图所示，∠BPA=126°，∠MQF=100°。求∠MAP的大小减去∠QFM的大小。
Construction (CDL):
Shape(CP,PB)
Shape(BP,PA)
Shape(AM,MF)
Shape(AP,PM,MA)
Shape(FM,MQ,QF)
Shape(FQ,QE)
Shape(QM,MP)
Collinear(BPMF)
Collinear(CPA)
Collinear(AMQE)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),126)
Equal(MeasureOfAngle(MQF),100)
Goal: Value(Sub(MeasureOfAngle(MAP),MeasureOfAngle(QFM)))
Answer: 46
Theorem sequence:
adjacent_complementary_angle(1,BPA,APF)
vertical_angle(1,PMA,FMQ)
triangle_property_angle_sum(1,FMQ)
triangle_property_angle_sum(1,APM)
Problem: As shown in the diagram, ∠DCE=30°, ∠GDE=25°, quadrilateral ABFG is a rectangle, triangle EDC is a right triangle. Find the measure of ∠BCD.
问题：如图所示，∠DCE=30°，∠GDE=25°，ABFG是长方形，三角形EDC为直角三角形。求∠BCD的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Shape(DC,CE,ED)
Shape(DE,EC,CF,FG,GD)
Collinear(ADG)
Collinear(BCF)
Conditions (CDL):
Equal(MeasureOfAngle(DCE),30)
Equal(MeasureOfAngle(GDE),25)
Rectangle(ABFG)
RightTriangle(EDC)
Goal: Value(MeasureOfAngle(BCD))
Answer: 115
Theorem sequence:
angle_addition(1,GDE,EDC)
parallel_property_collinear_extend(3,AG,BF,D)
parallel_property_collinear_extend(3,FB,GD,C)
parallel_property_alternate_interior_angle(1,CB,GD)
Problem: As shown in the diagram, AC=5, BC=12, AC and CB are the legs of the right triangle ACB. Find the sum of the length of line AC and the length of line AB.
问题：如图所示，AC=5，BC=12，∠ACB是直角三角形ACB的直角。求直线AC的长度与直线AB的长度之和。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),5)
Equal(LengthOfLine(BC),12)
RightTriangle(ACB)
Goal: Value(Add(LengthOfLine(AC),LengthOfLine(AB)))
Answer: 18
Theorem sequence:
right_triangle_property_pythagorean(1,ACB)
Problem: As shown in the diagram, ∠BGH=105°, AB is parallel to CD. Find the measure of ∠FHC.
问题：如图所示，∠BGH=105°，AB∥CD。求∠FHC的大小。
Construction (CDL):
Shape(AG,GE)
Shape(EG,GB)
Shape(BG,GH)
Shape(HG,GA)
Shape(CH,HG)
Shape(GH,HD)
Shape(DH,HF)
Shape(FH,HC)
Collinear(AGB)
Collinear(CHD)
Collinear(EGHF)
Conditions (CDL):
Equal(MeasureOfAngle(BGH),105)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(FHC))
Answer: 75
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,G)
parallel_property_collinear_extend(3,DC,BG,H)
parallel_property_ipsilateral_internal_angle(1,GB,HD)
vertical_angle(1,EHD,FHC)
Problem: As shown in the diagram, Add(LengthOfLine(AC)=LengthOfLine(BD)), the perimeter of triangle ABO is 18, E bisects segment AO, F bisects segment BO, BA and CD are opposite sides of the ▱ ABCD. Find the length of line EF.
问题：如图所示，Add(LengthOfLine(AC)=LengthOfLine(BD))，三角形ABO的周长为18，E是线段AO的中点，F是线段BO的中点，ABCD是平行四边形。求直线EF的长度。
Construction (CDL):
Shape(AB,BF,FE,EA)
Shape(OE,EF,FO)
Shape(AE,EO,OD,DA)
Shape(OF,FB,BC,CO)
Shape(DO,OC,CD)
Collinear(AEOC)
Collinear(BFOD)
Conditions (CDL):
Equal(Add(LengthOfLine(AC),LengthOfLine(BD)),24)
Equal(PerimeterOfTriangle(ABO),18)
IsMidpointOfLine(E,AO)
IsMidpointOfLine(F,BO)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(EF))
Answer: 3
Theorem sequence:
triangle_perimeter_formula(1,ABO)
line_addition(1,AO,OC)
line_addition(1,BO,OD)
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallelogram_property_diagonal_bisection(1,BCDA,O)
midsegment_of_triangle_judgment_midpoint(1,EF,OAB)
midsegment_of_triangle_property_length(1,EF,OAB)
Problem: As shown in the diagram, ∠ABC=118°, ∠DAC=40°, △ABC is mirror congruent to △ACD. Find the measure of ∠BCD.
问题：如图所示，∠ABC=118°，∠DAC=40°，三角形ABC镜像全等于三角形ACD。求∠BCD的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),118)
Equal(MeasureOfAngle(DAC),40)
MirrorCongruentBetweenTriangle(ABC,ACD)
Goal: Value(MeasureOfAngle(BCD))
Answer: 44
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,ABC,ACD)
mirror_congruent_triangle_property_angle_equal(1,BCA,DAC)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,ACD)
angle_addition(1,BCA,ACD)
Problem: As shown in the diagram, ∠ABC=34°, BC is parallel to DE, CA⊥BA. Find the measure of ∠FED.
问题：如图所示，∠ABC=34°，BC∥DE，CA⊥BA。求∠FED的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(CB,BD,DE,EC)
Shape(FE,ED)
Collinear(ABD)
Collinear(ACEF)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),34)
ParallelBetweenLine(BC,DE)
PerpendicularBetweenLine(CA,BA)
Goal: Value(MeasureOfAngle(FED))
Answer: 124
Theorem sequence:
triangle_property_angle_sum(1,ABC)
parallel_property_corresponding_angle(2,ED,CB,A)
adjacent_complementary_angle(1,FED,DEA)
Problem: As shown in the diagram, ∠GFB=42°, BD⊥ED, GF is perpendicular to DF. Find the measure of ∠CBD.
问题：如图所示，∠GFB=42°，BD⊥ED，GF垂直于DF。求∠CBD的大小。
Construction (CDL):
Shape(AB,BC)
Shape(CB,BD)
Shape(BD,DE)
Shape(DB,BF,FD)
Shape(GF,FB)
Collinear(ABD)
Collinear(EDF)
Collinear(CBF)
Conditions (CDL):
Equal(MeasureOfAngle(GFB),42)
PerpendicularBetweenLine(BD,ED)
PerpendicularBetweenLine(GF,DF)
Goal: Value(MeasureOfAngle(CBD))
Answer: 138
Theorem sequence:
adjacent_complementary_angle(1,FDB,BDE)
adjacent_complementary_angle(1,CBD,DBF)
perpendicular_judgment_angle(1,FD,BD)
parallel_judgment_per_per(1,FG,DB)
parallel_property_alternate_interior_angle(1,FG,DB)
Problem: As shown in the diagram, ∠ABO=30°, ∠ODA=20°, the center of ⊙O is O. Find the measure of ∠DAB.
问题：如图所示，∠ABO=30°，∠ODA=20°，O是⊙O的圆心。求∠DAB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AO,OD)
Shape(OA,AB,BO)
Shape(DO,OB,OBD)
Shape(BA,OAB)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),30)
Equal(MeasureOfAngle(ODA),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 50
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,OAB)
angle_addition(1,DAO,OAB)
Problem: As shown in the diagram, ∠FDG=135°, AB is parallel to EF. Find the measure of ∠DBA.
问题：如图所示，∠FDG=135°，AB∥EF。求∠DBA的大小。
Construction (CDL):
Shape(AB,BC)
Shape(DB,BA)
Shape(ED,DB)
Shape(BD,DF)
Shape(FD,DG)
Shape(GD,DE)
Collinear(CBDG)
Collinear(EDF)
Conditions (CDL):
Equal(MeasureOfAngle(FDG),135)
ParallelBetweenLine(AB,EF)
Goal: Value(MeasureOfAngle(DBA))
Answer: 45
Theorem sequence:
parallel_property_collinear_extend(3,FE,BA,D)
vertical_angle(1,EDC,FDG)
parallel_property_ipsilateral_internal_angle(1,DE,BA)
Problem: As shown in the diagram, ∠BDF=15°, ∠DBC=45°, AB∥DF. Find the measure of ∠ABC.
问题：如图所示，∠BDF=15°，∠DBC=45°，AB平行于DF。求∠ABC的大小。
Construction (CDL):
Shape(AB,BC)
Shape(DB,BA)
Shape(BD,DF)
Conditions (CDL):
Equal(MeasureOfAngle(BDF),15)
Equal(MeasureOfAngle(DBC),45)
ParallelBetweenLine(AB,DF)
Goal: Value(MeasureOfAngle(ABC))
Answer: 30
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,DF)
angle_addition(1,DBA,ABC)
Problem: As shown in the diagram, ∠AOC=95°, ∠DOF=53°. Find the measure of ∠EOB.
问题：如图所示，∠AOC=95°，∠DOF=53°。求∠EOB的大小。
Construction (CDL):
Shape(AO,OC)
Shape(CO,OE)
Shape(EO,OB)
Shape(BO,OD)
Shape(DO,OF)
Shape(FO,OA)
Collinear(AOB)
Collinear(COD)
Collinear(FOE)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),95)
Equal(MeasureOfAngle(DOF),53)
Goal: Value(MeasureOfAngle(EOB))
Answer: 32
Theorem sequence:
vertical_angle(1,COE,DOF)
angle_addition(1,AOC,COE)
adjacent_complementary_angle(1,AOE,EOB)
Problem: As shown in the diagram, CB=4, DB=7, D is the midpoint of segment AC. Find the length of line AB.
问题：如图所示，CB=4，DB=7，D平分线段AC。求直线AB的长度。
Construction (CDL):
Collinear(ADCB)
Conditions (CDL):
Equal(LengthOfLine(CB),4)
Equal(LengthOfLine(DB),7)
IsMidpointOfLine(D,AC)
Goal: Value(LengthOfLine(AB))
Answer: 10
Theorem sequence:
line_addition(1,AD,DB)
line_addition(1,DC,CB)
Problem: As shown in the diagram, ∠BOD=48°, the center of circle O is O, OB is parallel to AC. Find the measure of ∠OAB.
问题：如图所示，∠BOD=48°，O是圆O的圆心，OB平行于AC。求∠OAB的大小。
Construction (CDL):
Shape(AD,DO,OA)
Shape(OD,DB,BO)
Shape(AC,CD,DA)
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(BD,DC,OCB)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOD),48)
IsCentreOfCircle(O,O)
ParallelBetweenLine(OB,AC)
Goal: Value(MeasureOfAngle(OAB))
Answer: 24
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
parallel_property_alternate_interior_angle(1,OB,AC)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,ADO)
triangle_property_angle_sum(1,ODB)
triangle_property_angle_sum(1,OAC)
adjacent_complementary_angle(1,ADO,ODB)
Problem: As shown in the diagram, △ODC is congruent to △OBA, ∠AOB=15°, ∠COA=45°. Find the measure of ∠DOA.
问题：如图所示，△ODC与△OBA是全等三角形，∠AOB=15°，∠COA=45°。求∠DOA的大小。
Construction (CDL):
Shape(AO,OB,BA)
Shape(CO,OD,DC)
Shape(DO,OA)
Conditions (CDL):
CongruentBetweenTriangle(ODC,OBA)
Equal(MeasureOfAngle(AOB),15)
Equal(MeasureOfAngle(COA),45)
Goal: Value(MeasureOfAngle(DOA))
Answer: 30
Theorem sequence:
angle_addition(1,COD,DOA)
congruent_triangle_property_angle_equal(1,ODC,OBA)
Problem: As shown in the diagram, ∠OBA=33°, AB is parallel to CD, ∠AOB is the right angle of the right triangle AOB. Find the measure of ∠COA.
问题：如图所示，∠OBA=33°，AB平行于CD，三角形AOB为直角三角形。求∠COA的大小。
Construction (CDL):
Shape(CO,OA)
Shape(AO,OB,BA)
Shape(BO,OD)
Collinear(COD)
Conditions (CDL):
Equal(MeasureOfAngle(OBA),33)
ParallelBetweenLine(AB,CD)
RightTriangle(AOB)
Goal: Value(MeasureOfAngle(COA))
Answer: 57
Theorem sequence:
triangle_property_angle_sum(1,AOB)
parallel_property_collinear_extend(3,DC,BA,O)
parallel_property_alternate_interior_angle(1,OC,BA)
Problem: As shown in the diagram, ∠BCD=50°, AB⊥CB, BE is perpendicular to FE. Find the measure of ∠GFE.
问题：如图所示，∠BCD=50°，AB⊥CB，BE⊥FE。求∠GFE的大小。
Construction (CDL):
Shape(AB,BC)
Shape(BC,CD)
Shape(CB,BE,EF,FC)
Shape(GF,FE)
Collinear(DCFG)
Collinear(ABE)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),50)
PerpendicularBetweenLine(AB,CB)
PerpendicularBetweenLine(BE,FE)
Goal: Value(MeasureOfAngle(GFE))
Answer: 130
Theorem sequence:
adjacent_complementary_angle(1,FCB,BCD)
adjacent_complementary_angle(1,ABC,CBE)
perpendicular_judgment_angle(1,CB,EB)
parallel_judgment_per_per(1,BC,EF)
parallel_property_corresponding_angle(1,FE,CB,G)
Problem: As shown in the diagram, ∠CDA=78°, ∠DAC=∠BCA. Find the measure of ∠BCD.
问题：如图所示，∠CDA=78°，∠DAC=∠BCA。求∠BCD的大小。
Construction (CDL):
Shape(AC,CD,DA)
Shape(CA,AB,BC)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),78)
Equal(MeasureOfAngle(DAC),MeasureOfAngle(BCA))
Goal: Value(MeasureOfAngle(BCD))
Answer: 102
Theorem sequence:
triangle_property_angle_sum(1,ACD)
angle_addition(1,BCA,ACD)
Problem: As shown in the diagram, AB=10, AC=3, C is the midpoint of segment AD. Find the length of line DB.
问题：如图所示，AB=10，AC=3，C平分线段AD。求直线DB的长度。
Construction (CDL):
Collinear(ACDB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),3)
IsMidpointOfLine(C,AD)
Goal: Value(LengthOfLine(DB))
Answer: 4
Theorem sequence:
line_addition(1,AC,CD)
line_addition(1,AD,DB)
Problem: As shown in the diagram, ∠CEA=80°, ∠EAB=130°, BA is parallel to FE, FE is parallel to CD. Find the measure of ∠ECD.
问题：如图所示，∠CEA=80°，∠EAB=130°，BA平行于FE，FE平行于CD。求∠ECD的大小。
Construction (CDL):
Shape(EA,AB)
Shape(FE,EA)
Shape(EC,CD)
Shape(CE,EF)
Conditions (CDL):
Equal(MeasureOfAngle(CEA),80)
Equal(MeasureOfAngle(EAB),130)
ParallelBetweenLine(BA,FE)
ParallelBetweenLine(FE,CD)
Goal: Value(MeasureOfAngle(ECD))
Answer: 30
Theorem sequence:
angle_addition(1,CEF,FEA)
parallel_property_ipsilateral_internal_angle(1,EF,AB)
parallel_property_alternate_interior_angle(2,FE,CD)
Problem: As shown in the diagram, ∠ADE=90°, ∠CAB=90°, ∠DCF=30°, ∠FEA=45°, AE∥BC. Find the measure of ∠DFA.
问题：如图所示，∠ADE=90°，∠CAB=90°，∠DCF=30°，∠FEA=45°，AE∥BC。求∠DFA的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DF,FA)
Shape(FD,DC,CF)
Shape(AF,FE,EA)
Collinear(BDC)
Collinear(DFE)
Collinear(AFC)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),90)
Equal(MeasureOfAngle(CAB),90)
Equal(MeasureOfAngle(DCF),30)
Equal(MeasureOfAngle(FEA),45)
ParallelBetweenLine(AE,BC)
Goal: Value(MeasureOfAngle(DFA))
Answer: 75
Theorem sequence:
parallel_property_alternate_interior_angle(1,AE,BC)
triangle_property_angle_sum(1,AFE)
adjacent_complementary_angle(1,DFA,AFE)
Problem: As shown in the diagram, ∠GEF=20°, CF∥AE, AE is perpendicular to FE. Find the measure of ∠CGH.
问题：如图所示，∠GEF=20°，CF平行于AE，AE⊥FE。求∠CGH的大小。
Construction (CDL):
Shape(CG,GH)
Shape(HG,GF)
Shape(GE,EF,FG)
Shape(AE,EG)
Shape(EG,GC)
Collinear(HGE)
Collinear(CGF)
Conditions (CDL):
Equal(MeasureOfAngle(GEF),20)
ParallelBetweenLine(CF,AE)
PerpendicularBetweenLine(AE,FE)
Goal: Value(MeasureOfAngle(CGH))
Answer: 70
Theorem sequence:
parallel_property_collinear_extend(3,CF,AE,G)
angle_addition(1,AEG,GEF)
parallel_property_corresponding_angle(2,EA,GC,H)
Problem: As shown in the diagram, AC=4, the perimeter of △ADC is 12, DE perpendicular bisects AB. Find the length of line BC.
问题：如图所示，AC=4，△ADC的周长为12，DE垂直平分AB。求直线BC的长度。
Construction (CDL):
Shape(CA,AD,DC)
Shape(DA,AE,ED)
Shape(DE,EB,BD)
Collinear(CDB)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(PerimeterOfTriangle(ADC),12)
IsPerpendicularBisectorOfLine(DE,AB)
Goal: Value(LengthOfLine(BC))
Answer: 8
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,DE,AB)
triangle_perimeter_formula(1,CAD)
line_addition(1,CD,DB)
Problem: As shown in the diagram, Add(MeasureOfAngle(ABD)=MeasureOfAngle(DCA)), ∠CDB=120°. Find the measure of ∠CAB.
问题：如图所示，Add(MeasureOfAngle(ABD)=MeasureOfAngle(DCA))，∠CDB=120°。求∠CAB的大小。
Construction (CDL):
Shape(AB,BD,DC,CA)
Shape(DB,BC,CD)
Conditions (CDL):
Equal(Add(MeasureOfAngle(ABD),MeasureOfAngle(DCA)),55)
Equal(MeasureOfAngle(CDB),120)
Goal: Value(MeasureOfAngle(CAB))
Answer: 65
Theorem sequence:
angle_addition(1,ABD,DBC)
angle_addition(1,BCD,DCA)
triangle_property_angle_sum(1,DBC)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠CBD=130°. Find the sum of the measure of ∠ABC and the measure of ∠DBE.
问题：如图所示，∠CBD=130°。求∠ABC的大小与∠DBE的大小之和。
Construction (CDL):
Shape(AB,BC)
Shape(CB,BD)
Shape(DB,BE)
Shape(EB,BA)
Collinear(ABD)
Collinear(CBE)
Conditions (CDL):
Equal(MeasureOfAngle(CBD),130)
Goal: Value(Add(MeasureOfAngle(ABC),MeasureOfAngle(DBE)))
Answer: 100
Theorem sequence:
adjacent_complementary_angle(1,ABC,CBD)
adjacent_complementary_angle(1,CBD,DBE)
Problem: As shown in the diagram, ∠ABC=65°, ∠ACD=40°, DC∥AB. Find the measure of ∠BCA.
问题：如图所示，∠ABC=65°，∠ACD=40°，DC平行于AB。求∠BCA的大小。
Construction (CDL):
Shape(AC,CD)
Shape(CA,AB,BC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),65)
Equal(MeasureOfAngle(ACD),40)
ParallelBetweenLine(DC,AB)
Goal: Value(MeasureOfAngle(BCA))
Answer: 75
Theorem sequence:
angle_addition(1,BCA,ACD)
parallel_property_ipsilateral_internal_angle(1,BA,CD)
Problem: As shown in the diagram, ∠BCA=43°, ∠HGA=90°, BC is parallel to MG, MG∥EF. Find the measure of ∠HFE.
问题：如图所示，∠BCA=43°，∠HGA=90°，BC∥MG，MG∥EF。求∠HFE的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BM,MG,GC,CB)
Shape(ME,EF,FG,GM)
Shape(FE,EH,HF)
Collinear(ABMEH)
Collinear(ACG)
Collinear(HFG)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),43)
Equal(MeasureOfAngle(HGA),90)
ParallelBetweenLine(BC,MG)
ParallelBetweenLine(MG,EF)
Goal: Value(MeasureOfAngle(HFE))
Answer: 47
Theorem sequence:
angle_addition(1,FGM,MGC)
parallel_property_corresponding_angle(1,FE,GM,H)
parallel_property_corresponding_angle(2,GM,CB,A)
Problem: As shown in the diagram, ∠ABC=60°, ∠DEA=40°, DE∥BC. Find the measure of ∠EAD.
问题：如图所示，∠ABC=60°，∠DEA=40°，DE∥BC。求∠EAD的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),60)
Equal(MeasureOfAngle(DEA),40)
ParallelBetweenLine(DE,BC)
Goal: Value(MeasureOfAngle(EAD))
Answer: 80
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
triangle_property_angle_sum(1,ADE)
Problem: As shown in the diagram, ∠BAC=25°, the center of circle O is O, MA is the tangent to circle O, MB is the tangent to circle O. Find the measure of ∠BMA.
问题：如图所示，∠BAC=25°，⊙O的圆心为O，MA是⊙O的切线，MB是⊙O的切线。求∠BMA的大小。
Construction (CDL):
Shape(BM,MA,OBA)
Shape(AB,OBA)
Shape(BA,AO,OC,OCB)
Shape(CO,OA,OAC)
Collinear(COA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),25)
IsCentreOfCircle(O,O)
IsTangentOfCircle(MA,O)
IsTangentOfCircle(MB,O)
Goal: Value(MeasureOfAngle(BMA))
Answer: 50
Theorem sequence:
angle_addition(1,MAB,BAC)
tangent_of_circle_property_perpendicular(2,MA,O,O)
tangent_of_circle_property_length_equal(1,MB,MA,O)
isosceles_triangle_judgment_line_equal(1,MAB)
isosceles_triangle_property_angle_equal(1,MAB)
triangle_property_angle_sum(1,BMA)