SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠AOC=140°, the center of circle O is O, B is the midpoint of ⌒OCA. Find the measure of ∠ADB.
问题：如图所示，∠AOC=140°，圆O的圆心为O，B是⌒OCA的中点。求∠ADB的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(EA,AD,DE)
Shape(OBA,AE,EB)
Shape(OE,ED,ODC,CO)
Shape(EO,OC,OCB,BE)
Collinear(DEB)
Collinear(AEO)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),140)
IsCentreOfCircle(O,O)
IsMidpointOfArc(B,OCA)
Goal: Value(MeasureOfAngle(ADB))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_addition_length(1,OCB,OBA)
similar_arc_judgment_cocircular(1,OCA,OBA)
similar_arc_property_length_ratio(1,OCA,OBA)
similar_arc_property_measure_ratio(1,OCA,OBA)
arc_property_circumference_angle_external(1,OBA,D)
Problem: As shown in the diagram, ∠DOB=90°, the center of circle O is O. Find the measure of ∠BCD.
问题：如图所示，∠DOB=90°，O是圆O的圆心。求∠BCD的大小。
Construction (CDL):
Shape(OCD,DC)
Shape(OBC,CB)
Shape(CD,DO,OB,BC)
Shape(OD,ODB,BO)
Cocircular(O,DBC)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),90)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 135
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠DBA=50°, the diameter of ⊙O is AB. Find the measure of ∠BCD.
问题：如图所示，∠DBA=50°，AB是圆O的直径。求∠BCD的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(OBC,CB)
Shape(EA,AD,DE)
Shape(ED,DB,BE)
Shape(OAD,DA)
Shape(ODB,BD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(DBA),50)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,C)
diameter_of_circle_property_right_angle(1,BCA,O)
angle_addition(1,BCD,DCA)
Problem: As shown in the diagram, BC=10, BD is the angle bisector of ∠ABC, ∠ABC and ∠BCA are the base angles of the isosceles right triangle ABC, DE⊥CE. Find the perimeter of △DEC.
问题：如图所示，BC=10，BD平分∠ABC，△ABC为等腰直角△，DE垂直于CE。求△DEC的周长。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DB,BE,ED)
Shape(DE,EC,CD)
Collinear(ADC)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(BC),10)
IsBisectorOfAngle(BD,ABC)
IsoscelesRightTriangle(ABC)
PerpendicularBetweenLine(DE,CE)
Goal: Value(PerimeterOfTriangle(DEC))
Answer: 10
Theorem sequence:
adjacent_complementary_angle(1,BED,DEC)
mirror_congruent_triangle_judgment_aas(1,DBE,DAB)
mirror_congruent_triangle_property_line_equal(1,DBE,DAB)
mirror_congruent_triangle_property_line_equal(1,BED,BDA)
line_addition(1,AD,DC)
line_addition(1,BE,EC)
triangle_perimeter_formula(1,DEC)
Problem: As shown in the diagram, Add(MeasureOfAngle(DBA)=MeasureOfAngle(ACE)). Find the measure of ∠CAB.
问题：如图所示，Add(MeasureOfAngle(DBA)=MeasureOfAngle(ACE))。求∠CAB的大小。
Construction (CDL):
Shape(DB,BA)
Shape(AB,BC,CA)
Shape(AC,CE)
Collinear(DBCE)
Conditions (CDL):
Equal(Add(MeasureOfAngle(DBA),MeasureOfAngle(ACE)),230)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
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, ∠EBF=32°, triangle EBF is mirror congruent to triangle EFD. Find the measure of ∠AEG minus the measure of ∠GFD.
问题：如图所示，∠EBF=32°，△EBF与△EFD是镜像全等三角形。求∠AEG的大小减去∠GFD的大小。
Construction (CDL):
Shape(EB,BF,FE)
Shape(EF,FG,GE)
Shape(AE,EG,GC,CA)
Shape(GF,FD,DG)
Collinear(AEB)
Collinear(BFGC)
Collinear(EGD)
Conditions (CDL):
Equal(MeasureOfAngle(EBF),32)
MirrorCongruentBetweenTriangle(EBF,EFD)
Goal: Value(Sub(MeasureOfAngle(AEG),MeasureOfAngle(GFD)))
Answer: 64
Theorem sequence:
adjacent_complementary_angle(1,AEG,GEB)
adjacent_complementary_angle(1,DGF,FGE)
mirror_congruent_triangle_property_angle_equal(1,BFE,DEF)
triangle_property_angle_sum(1,GFD)
triangle_property_angle_sum(1,EBG)
Problem: As shown in the diagram, ∠BAC=40°, ∠CBA=60°, DF is parallel to AC. Find the measure of ∠DFB.
问题：如图所示，∠BAC=40°，∠CBA=60°，DF平行于AC。求∠DFB的大小。
Construction (CDL):
Shape(BD,DF,FB)
Shape(DA,AC,CF,FD)
Collinear(BDA)
Collinear(BFC)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),40)
Equal(MeasureOfAngle(CBA),60)
ParallelBetweenLine(DF,AC)
Goal: Value(MeasureOfAngle(DFB))
Answer: 80
Theorem sequence:
parallel_property_corresponding_angle(1,DF,AC,B)
triangle_property_angle_sum(1,BDF)
Problem: As shown in the diagram, ∠BCD=110°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠AOD.
问题：如图所示，∠BCD=110°，O是圆O的圆心，圆O的直径为AB。求∠AOD的大小。
Construction (CDL):
Shape(OCD,DC)
Shape(OD,ODA,AO)
Shape(CD,DO,OB,BC)
Shape(OBC,CB)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),110)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(AOD))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_internal(1,OBD,C)
arc_property_center_angle(1,OBD,O)
adjacent_complementary_angle(1,AOD,DOB)
Problem: As shown in the diagram, ∠BAC=68°, AB is parallel to CD. Find the measure of ∠AEF minus the measure of ∠CFE.
问题：如图所示，∠BAC=68°，AB∥CD。求∠AEF的大小减去∠CFE的大小。
Construction (CDL):
Shape(BA,AE,EF,FD,DB)
Shape(FE,EC,CF)
Collinear(AEC)
Collinear(CFD)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),68)
ParallelBetweenLine(AB,CD)
Goal: Value(Sub(MeasureOfAngle(AEF),MeasureOfAngle(CFE)))
Answer: 112
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AB,CD)
adjacent_complementary_angle(1,AEF,FEC)
triangle_property_angle_sum(1,FEC)
Problem: As shown in the diagram, ∠COM=46°, ∠NOB=22°. Find the measure of ∠AOD.
问题：如图所示，∠COM=46°，∠NOB=22°。求∠AOD的大小。
Construction (CDL):
Shape(MO,OA)
Shape(AO,OD)
Shape(DO,ON)
Shape(NO,OB)
Shape(BO,OC)
Shape(CO,OM)
Collinear(MON)
Collinear(AOB)
Collinear(COD)
Conditions (CDL):
Equal(MeasureOfAngle(COM),46)
Equal(MeasureOfAngle(NOB),22)
Goal: Value(MeasureOfAngle(AOD))
Answer: 112
Theorem sequence:
vertical_angle(1,COM,DON)
angle_addition(1,DON,NOB)
adjacent_complementary_angle(1,AOD,DOB)
Problem: As shown in the diagram, ∠DOA=136°, MO is perpendicular to BO. Find the measure of ∠COM.
问题：如图所示，∠DOA=136°，MO⊥BO。求∠COM的大小。
Construction (CDL):
Shape(AO,OC)
Shape(CO,OM)
Shape(MO,OB)
Shape(BO,OD)
Shape(DO,OA)
Collinear(AOB)
Collinear(COD)
Conditions (CDL):
Equal(MeasureOfAngle(DOA),136)
PerpendicularBetweenLine(MO,BO)
Goal: Value(MeasureOfAngle(COM))
Answer: 46
Theorem sequence:
vertical_angle(1,COB,DOA)
angle_addition(1,COM,MOB)
Problem: As shown in the diagram, AB=AC, ∠ADE=55°, DE∥BC. Find the measure of ∠BCA.
问题：如图所示，AB=AC，∠ADE=55°，DE∥BC。求∠BCA的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(ADE),55)
ParallelBetweenLine(DE,BC)
Goal: Value(MeasureOfAngle(BCA))
Answer: 55
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
Problem: As shown in the diagram, CA=CD, DC=DB, ∠CAD=40°. Find the measure of ∠DBC.
问题：如图所示，CA=CD，DC=DB，∠CAD=40°。求∠DBC的大小。
Construction (CDL):
Shape(CA,AD,DC)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(CA),LengthOfLine(CD))
Equal(LengthOfLine(DC),LengthOfLine(DB))
Equal(MeasureOfAngle(CAD),40)
Goal: Value(MeasureOfAngle(DBC))
Answer: 20
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,CAD)
isosceles_triangle_property_angle_equal(1,CAD)
adjacent_complementary_angle(1,ADC,CDB)
isosceles_triangle_judgment_line_equal(1,DBC)
isosceles_triangle_property_angle_equal(1,DBC)
triangle_property_angle_sum(1,DBC)
Problem: As shown in the diagram, ∠CAB=66°, O is the center of ⊙O. Find the measure of ∠BCO.
问题：如图所示，∠CAB=66°，⊙O的圆心为O。求∠BCO的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OC,CA,AB,BO)
Shape(CO,OB,BC)
Shape(OBC,CB)
Shape(OAB,BA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),66)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 24
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,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, AB=AC, ∠CAB=50°, DE is the perpendicular bisector of CA. Find the measure of ∠BCD.
问题：如图所示，AB=AC，∠CAB=50°，DE垂直平分AB。求∠BCD的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Shape(DB,BC,CD)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),50)
IsPerpendicularBisectorOfLine(DE,CA)
Goal: Value(MeasureOfAngle(BCD))
Answer: 15
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
perpendicular_bisector_property_distance_equal(1,DE,CA)
isosceles_triangle_judgment_line_equal(1,DCA)
isosceles_triangle_property_angle_equal(1,DCA)
angle_addition(1,BCD,DCA)
Problem: As shown in the diagram, ∠DAB=110°, BA is parallel to CD. Find the measure of ∠ADE.
问题：如图所示，∠DAB=110°，BA平行于CD。求∠ADE的大小。
Construction (CDL):
Shape(BC,CD,DA,AB)
Shape(AD,DE)
Collinear(CDE)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),110)
ParallelBetweenLine(BA,CD)
Goal: Value(MeasureOfAngle(ADE))
Answer: 110
Theorem sequence:
parallel_property_collinear_extend(1,DC,AB,E)
parallel_property_alternate_interior_angle(2,ED,AB)
Problem: As shown in the diagram, triangle AED is congruent to triangle ABC, ∠CAB=30°, ∠DAC=a°, ∠EAB=a°, DC is parallel to AB. Find the value of a.
问题：如图所示，△AED全等于△ABC，∠CAB=30°，∠DAC=a°，∠EAB=a°，DC∥AB。求a的值。
Construction (CDL):
Shape(ED,DF,FE)
Shape(FD,DA,AF)
Shape(CF,FA,AC)
Shape(CA,AB,BC)
Collinear(EFA)
Collinear(CFD)
Conditions (CDL):
CongruentBetweenTriangle(AED,ABC)
Equal(MeasureOfAngle(CAB),30)
Equal(MeasureOfAngle(DAC),a)
Equal(MeasureOfAngle(EAB),a)
ParallelBetweenLine(DC,AB)
Goal: Value(a)
Answer: 120
Theorem sequence:
congruent_triangle_property_line_equal(1,EDA,BCA)
parallel_property_alternate_interior_angle(2,DC,AB)
isosceles_triangle_judgment_line_equal(1,ACD)
isosceles_triangle_property_angle_equal(1,ACD)
triangle_property_angle_sum(1,ACD)
Problem: As shown in the diagram, ∠CAB=40°, ED is the perpendicular bisector of AB, AB and AC are the legs of the isosceles △ ABC. Find the measure of ∠EBC.
问题：如图所示，∠CAB=40°，ED垂直平分AB，AB和AC是等腰△ABC的腰。求∠EBC的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DB,BE)
Shape(EB,BC,CE)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsPerpendicularBisectorOfLine(ED,AB)
IsoscelesTriangle(ABC)
Goal: Value(MeasureOfAngle(EBC))
Answer: 30
Theorem sequence:
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
perpendicular_bisector_property_distance_equal(1,ED,AB)
isosceles_triangle_judgment_line_equal(1,EAB)
isosceles_triangle_property_angle_equal(1,EAB)
angle_addition(1,ABE,EBC)
Problem: As shown in the diagram, AB=13, AC=5, ∠BCA=90°, DE perpendicular bisects AB. Find the perimeter of △ADC.
问题：如图所示，AB=13，AC=5，∠BCA=90°，DE是AB的垂直平分线。求三角形ADC的周长。
Construction (CDL):
Shape(CA,AD,DC)
Shape(DA,AE,ED)
Shape(DE,EB,BD)
Collinear(CDB)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),13)
Equal(LengthOfLine(AC),5)
Equal(MeasureOfAngle(BCA),90)
IsPerpendicularBisectorOfLine(DE,AB)
Goal: Value(PerimeterOfTriangle(ADC))
Answer: 17
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
perpendicular_bisector_property_distance_equal(1,DE,AB)
line_addition(1,CD,DB)
triangle_perimeter_formula(1,ADC)
Problem: As shown in the diagram, ∠CAB=76°, AD is the angle bisector of ∠CAB, AE bisects ∠DAB. Find the measure of ∠DAE.
问题：如图所示，∠CAB=76°，AD是∠CAB的角平分线，AE平分∠DAB。求∠DAE的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,ED,DA)
Shape(AD,DC,CA)
Collinear(BEDC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),76)
IsBisectorOfAngle(AD,CAB)
IsBisectorOfAngle(AE,DAB)
Goal: Value(MeasureOfAngle(DAE))
Answer: 19
Theorem sequence:
angle_addition(1,CAD,DAB)
angle_addition(1,DAE,EAB)
Problem: As shown in the diagram, ∠DAB=130°, BD bisects ∠ABC, AD is parallel to BC. Find the measure of ∠CDB.
问题：如图所示，∠DAB=130°，BD是∠ABC的角平分线，AD∥BC。求∠CDB的大小。
Construction (CDL):
Shape(OCD,DC)
Shape(ODA,AD)
Shape(OAB,BA)
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Shape(CB,OBC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),130)
IsBisectorOfAngle(BD,ABC)
ParallelBetweenLine(AD,BC)
Goal: Value(MeasureOfAngle(CDB))
Answer: 105
Theorem sequence:
parallel_property_alternate_interior_angle(2,AD,BC)
triangle_property_angle_sum(1,ABD)
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_internal(1,ODB,A)
triangle_property_angle_sum(1,DBC)
Problem: As shown in the diagram, ∠ECD=25°, ∠FEA=∠DEC, ABCD is a rectangle. Find the measure of ∠FEA.
问题：如图所示，∠ECD=25°，∠FEA=∠DEC，ABCD是矩形。求∠FEA的大小。
Construction (CDL):
Shape(AB,BF,FE,EA)
Shape(EF,FC,CE)
Shape(EC,CD,DE)
Collinear(AED)
Collinear(BFC)
Conditions (CDL):
Equal(MeasureOfAngle(ECD),25)
Equal(MeasureOfAngle(FEA),MeasureOfAngle(DEC))
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(FEA))
Answer: 65
Theorem sequence:
triangle_property_angle_sum(1,ECD)
Problem: As shown in the diagram, ∠BAD=25°, the center of circle O is O, AB is the diameter of circle O, CD is the tangent to circle O. Find the measure of ∠DCA.
问题：如图所示，∠BAD=25°，圆O的圆心为O，AB是⊙O的直径，圆O的切线为CD。求∠DCA的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OA,AD,DO)
Shape(OD,ODB,BO)
Shape(OAD,DA)
Shape(DC,CB,ODB)
Collinear(AOBC)
Cocircular(O,ADB)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(CD,O)
Goal: Value(MeasureOfAngle(DCA))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,A)
arc_property_center_angle(1,ODB,O)
tangent_of_circle_property_perpendicular(1,CD,O,O)
triangle_property_angle_sum(1,ODC)
Problem: As shown in the diagram, ∠DAE=∠ABC, ∠EAC=75°. Find the measure of ∠BCA.
问题：如图所示，∠DAE=∠ABC，∠EAC=75°。求∠BCA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DA,AE)
Shape(EA,AC)
Shape(AC,CF)
Collinear(DAB)
Collinear(BCF)
Conditions (CDL):
Equal(MeasureOfAngle(DAE),MeasureOfAngle(ABC))
Equal(MeasureOfAngle(EAC),75)
Goal: Value(MeasureOfAngle(BCA))
Answer: 75
Theorem sequence:
triangle_property_angle_sum(1,ABC)
adjacent_complementary_angle(1,DAC,CAB)
angle_addition(1,DAE,EAC)
Problem: As shown in the diagram, ∠CDA=76°, quadrilateral ABCD is a parallelogram. Find the measure of ∠DAB.
问题：如图所示，∠CDA=76°，BA和CD是▱ABCD的一组对边。求∠DAB的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),76)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(DAB))
Answer: 104
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,DC,AB)
Problem: As shown in the diagram, ∠BNY=60°, ∠EMN=70°, ∠MXY=70°. Find the measure of ∠XYN.
问题：如图所示，∠BNY=60°，∠EMN=70°，∠MXY=70°。求∠XYN的大小。
Construction (CDL):
Shape(AM,ME)
Shape(EM,MN)
Shape(MN,NG)
Shape(GN,NB)
Shape(BN,NY)
Shape(BY,YD)
Shape(DY,YH)
Shape(HY,YX)
Shape(YX,XF)
Shape(FX,XC)
Shape(CX,XM)
Shape(XM,MA)
Shape(MX,XY,YN,NM)
Collinear(AMNB)
Collinear(CXYD)
Collinear(EMXF)
Collinear(GNYH)
Conditions (CDL):
Equal(MeasureOfAngle(BNY),60)
Equal(MeasureOfAngle(EMN),70)
Equal(MeasureOfAngle(MXY),70)
Goal: Value(MeasureOfAngle(XYN))
Answer: 60
Theorem sequence:
adjacent_complementary_angle(1,EMN,NMX)
adjacent_complementary_angle(1,BNY,YNM)
quadrilateral_property_angle_sum(1,MXYN)
Problem: As shown in the diagram, ∠BCA=40°, O is the center of circle O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=40°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OCA,AO,OD,DC)
Shape(DO,OA,AD)
Shape(CD,DB,BC)
Shape(DA,OAB,BD)
Shape(OBC,CB)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠COB=120°, the center of ⊙O is O. Find the measure of ∠CAB.
问题：如图所示，∠COB=120°，O是⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OCA,AC)
Shape(AB,BO,OC,CA)
Shape(OB,OBC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠AOC=40°, OD bisects ∠COB. Find the measure of ∠DOB.
问题：如图所示，∠AOC=40°，OD是∠COB的角平分线。求∠DOB的大小。
Construction (CDL):
Shape(AO,OC)
Shape(CO,OD)
Shape(DO,OB)
Collinear(AOB)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),40)
IsBisectorOfAngle(OD,COB)
Goal: Value(MeasureOfAngle(DOB))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,AOC,COB)
angle_addition(1,COD,DOB)
Problem: As shown in the diagram, ∠ABC=80°, AE is the angle bisector of ∠DAG, DA∥BC. Find the measure of ∠EAG.
问题：如图所示，∠ABC=80°，AE平分∠DAG，DA平行于BC。求∠EAG的大小。
Construction (CDL):
Shape(EA,AG)
Shape(DA,AE)
Shape(BA,AD)
Shape(AB,BC)
Collinear(GAB)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),80)
IsBisectorOfAngle(AE,DAG)
ParallelBetweenLine(DA,BC)
Goal: Value(MeasureOfAngle(EAG))
Answer: 50
Theorem sequence:
parallel_property_alternate_interior_angle(2,DA,BC)
adjacent_complementary_angle(1,BAD,DAG)
angle_addition(1,DAE,EAG)
Problem: As shown in the diagram, ∠ABC=140°, CD∥AB. Find the measure of ∠DCB.
问题：如图所示，∠ABC=140°，CD平行于AB。求∠DCB的大小。
Construction (CDL):
Shape(AB,BC)
Shape(DC,CB)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),140)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(DCB))
Answer: 140
Theorem sequence:
parallel_property_alternate_interior_angle(1,CD,AB)
Problem: As shown in the diagram, BD=120, DC=60, EC=50, AB is perpendicular to DB, EC⊥DC. Find the length of line AB.
问题：如图所示，BD=120，DC=60，EC=50，AB垂直于DB，EC⊥DC。求直线AB的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(CD,DE,EC)
Shape(AD,DC)
Shape(ED,DB)
Collinear(ADE)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(BD),120)
Equal(LengthOfLine(DC),60)
Equal(LengthOfLine(EC),50)
PerpendicularBetweenLine(AB,DB)
PerpendicularBetweenLine(EC,DC)
Goal: Value(LengthOfLine(AB))
Answer: 100
Theorem sequence:
vertical_angle(1,BDA,CDE)
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, ∠BOC=50°, O is the center of circle O, OE⊥BE. Find the measure of ∠CDA.
问题：如图所示，∠BOC=50°，⊙O的圆心为O，OE垂直于BE。求∠CDA的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(FD,DA,AF)
Shape(DF,FE,EO,OB,OBD)
Shape(FA,OAC,CF)
Shape(EF,FC,CE)
Shape(OE,EB,BO)
Shape(BE,EC,OCB)
Collinear(AFEB)
Collinear(DFC)
Collinear(OEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,BE)
Goal: Value(MeasureOfAngle(CDA))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OCB,O)
adjacent_complementary_angle(1,AEO,OEB)
circle_property_chord_perpendicular_bisect_arc(1,OAB,OEC)
congruent_arc_judgment_length_equal(1,OAC,OCB)
congruent_arc_property_measure_equal(1,OAC,OCB)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, △CAB is congruent to △EAD, ∠BAD=70°, ∠BCA=90°, ∠CAB=70°, ∠DEA=90°. Find the measure of ∠DBA.
问题：如图所示，△CAB全等于△EAD，∠BAD=70°，∠BCA=90°，∠CAB=70°，∠DEA=90°。求∠DBA的大小。
Construction (CDL):
Shape(BC,CA,AE,EB)
Shape(BE,ED,DB)
Shape(DE,EA,AD)
Collinear(BEA)
Conditions (CDL):
CongruentBetweenTriangle(CAB,EAD)
Equal(MeasureOfAngle(BAD),70)
Equal(MeasureOfAngle(BCA),90)
Equal(MeasureOfAngle(CAB),70)
Equal(MeasureOfAngle(DEA),90)
Goal: Value(MeasureOfAngle(DBA))
Answer: 55
Theorem sequence:
congruent_triangle_property_line_equal(1,CAB,EAD)
isosceles_triangle_judgment_line_equal(1,ADB)
isosceles_triangle_property_angle_equal(1,ADB)
triangle_property_angle_sum(1,ADB)
Problem: As shown in the diagram, △ABC is congruent to △AED, ∠CAB=30°, AF perpendicular bisects CD. Find the measure of ∠DAC.
问题：如图所示，△ABC全等于△AED，∠CAB=30°，AF是CD的垂直平分线。求∠DAC的大小。
Construction (CDL):
Shape(ED,DF,FE)
Shape(FD,DA,AF)
Shape(CF,FA,AC)
Shape(CA,AB,BC)
Collinear(DFC)
Collinear(EFA)
Conditions (CDL):
CongruentBetweenTriangle(ABC,AED)
Equal(MeasureOfAngle(CAB),30)
IsPerpendicularBisectorOfLine(AF,CD)
Goal: Value(MeasureOfAngle(DAC))
Answer: 60
Theorem sequence:
congruent_triangle_property_line_equal(1,BCA,EDA)
congruent_triangle_property_angle_equal(1,ABC,AED)
mirror_congruent_triangle_judgment_hl(1,AFD,ACF)
mirror_congruent_triangle_property_angle_equal(1,AFD,ACF)
angle_addition(1,DAF,FAC)
Problem: As shown in the diagram, ∠BDC=130°, the center of ⊙O is O. Find the measure of ∠COB.
问题：如图所示，∠BDC=130°，O是⊙O的圆心。求∠COB的大小。
Construction (CDL):
Shape(OCB,BO,OC)
Shape(OB,BD,DC,CO)
Shape(OBD,DB)
Shape(ODC,CD)
Cocircular(O,BDC)
Conditions (CDL):
Equal(MeasureOfAngle(BDC),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_internal(1,OBC,D)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, AB=AC, ∠CAB=90°, the perimeter of △DEC is 8, BD bisects ∠ABC, DE is perpendicular to CE. Find the length of line BC.
问题：如图所示，AB=AC，∠CAB=90°，三角形DEC的周长为8，BD平分∠ABC，DE⊥CE。求直线BC的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DB,BE,ED)
Shape(DE,EC,CD)
Collinear(BEC)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),90)
Equal(PerimeterOfTriangle(DEC),8)
IsBisectorOfAngle(BD,ABC)
PerpendicularBetweenLine(DE,CE)
Goal: Value(LengthOfLine(BC))
Answer: 8
Theorem sequence:
adjacent_complementary_angle(1,BED,DEC)
bisector_of_angle_judgment_angle_equal(1,BD,ABE)
bisector_of_angle_property_distance_equal(1,BD,ABE)
line_addition(1,AD,DC)
line_addition(1,BE,EC)
mirror_congruent_triangle_judgment_hl(1,BED,BDA)
mirror_congruent_triangle_property_line_equal(1,DBE,DAB)
triangle_perimeter_formula(1,DEC)
Problem: As shown in the diagram, ∠CAB=30°, ∠DCA=20°, △ABC is a right △. Find the measure of ∠CDB.
问题：如图所示，∠CAB=30°，∠DCA=20°，AB和BC是直角△ABC的两个直角边。求∠CDB的大小。
Construction (CDL):
Shape(AD,DC,CA)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),30)
Equal(MeasureOfAngle(DCA),20)
RightTriangle(ABC)
Goal: Value(MeasureOfAngle(CDB))
Answer: 50
Theorem sequence:
angle_addition(1,BCD,DCA)
triangle_property_angle_sum(1,CAB)
triangle_property_angle_sum(1,CDB)
Problem: As shown in the diagram, ∠CDB=110°, BD bisects ∠ABC, CD bisects ∠BCA. Find the measure of ∠CAB.
问题：如图所示，∠CDB=110°，BD平分∠ABC，CD平分∠BCA。求∠CAB的大小。
Construction (CDL):
Shape(DB,BC,CD)
Shape(AB,BD,DC,CA)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),110)
IsBisectorOfAngle(BD,ABC)
IsBisectorOfAngle(CD,BCA)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,DBC)
triangle_property_angle_sum(1,BCA)
angle_addition(1,BCD,DCA)
angle_addition(1,ABD,DBC)
Problem: As shown in the diagram, ∠ABE=32°, ∠CAB=70°, ∠DCA=20°. Find the measure of ∠EFC.
问题：如图所示，∠ABE=32°，∠CAB=70°，∠DCA=20°。求∠EFC的大小。
Construction (CDL):
Shape(CE,EF,FC)
Shape(CF,FB,BC)
Shape(EA,AD,DF,FE)
Shape(FD,DB,BF)
Collinear(CFD)
Collinear(EFB)
Collinear(CEA)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),32)
Equal(MeasureOfAngle(CAB),70)
Equal(MeasureOfAngle(DCA),20)
Goal: Value(MeasureOfAngle(EFC))
Answer: 58
Theorem sequence:
triangle_property_angle_sum(1,CEF)
triangle_property_angle_sum(1,EAB)
adjacent_complementary_angle(1,CEF,FEA)
Problem: As shown in the diagram, the diameter of circle O is 10, ED=8, the center of ⊙O is O, AD is perpendicular to OD. Find the length of line AB.
问题：如图所示，圆O的直径为10，ED=8，圆O的圆心为O，AD⊥OD。求直线AB的长度。
Construction (CDL):
Shape(AO,OE,OEA)
Shape(AD,DO,OA)
Shape(OD,DB,BO)
Shape(BA,OAB)
Shape(OB,OBE,EO)
Collinear(EOD)
Collinear(ADB)
Cocircular(O,ABE)
Conditions (CDL):
Equal(DiameterOfCircle(O),10)
Equal(LengthOfLine(ED),8)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AD,OD)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OE,O)
line_addition(1,EO,OD)
line_addition(1,AD,DB)
right_triangle_judgment_angle(1,ADO)
right_triangle_property_pythagorean(1,ADO)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,AB)