SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠AED=∠FEA, ∠CEF=50°, quadrilateral ABCD is a rectangle. Find the measure of ∠AED.
问题：如图所示，∠AED=∠FEA，∠CEF=50°，四边形ABCD是长方形。求∠AED的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(EA,AF,FE)
Shape(DF,FA,AB,BC,CE)
Collinear(DEC)
Conditions (CDL):
Equal(MeasureOfAngle(AED),MeasureOfAngle(FEA))
Equal(MeasureOfAngle(CEF),50)
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(AED))
Answer: 65
Theorem sequence:
angle_addition(1,FEA,AED)
adjacent_complementary_angle(1,CEF,FED)
Problem: As shown in the diagram, triangle AOB is congruent to triangle DOC, ∠AOC=100°, ∠AOD=30°, ∠EOC=30°. Find the measure of ∠DOB.
问题：如图所示，△AOB全等于△DOC，∠AOC=100°，∠AOD=30°，∠EOC=30°。求∠DOB的大小。
Construction (CDL):
Shape(AO,OD,DA)
Shape(DO,OE,ED)
Shape(DE,EB,BD)
Shape(EO,OC,CE)
Collinear(DEC)
Collinear(BEO)
Collinear(ADB)
Conditions (CDL):
CongruentBetweenTriangle(AOB,DOC)
Equal(MeasureOfAngle(AOC),100)
Equal(MeasureOfAngle(AOD),30)
Equal(MeasureOfAngle(EOC),30)
Goal: Value(MeasureOfAngle(DOB))
Answer: 40
Theorem sequence:
angle_addition(1,AOD,DOE)
angle_addition(1,AOE,EOC)
Problem: As shown in the diagram, BD=25, BF=30, CD=3/2, BA is parallel to DC, DC∥FE. Find the length of line AB.
问题：如图所示，BD=25，BF=30，CD=3/2，BA平行于DC，DC平行于FE。求直线AB的长度。
Construction (CDL):
Shape(AB,BD,DC,CA)
Shape(CD,DF,FC)
Shape(CF,FE)
Collinear(BDF)
Collinear(ACF)
Conditions (CDL):
Equal(LengthOfLine(BD),25)
Equal(LengthOfLine(BF),30)
Equal(LengthOfLine(CD),3/2)
ParallelBetweenLine(BA,DC)
ParallelBetweenLine(DC,FE)
Goal: Value(LengthOfLine(AB))
Answer: 9
Theorem sequence:
parallel_property_corresponding_angle(2,BA,DC,F)
similar_triangle_judgment_aa(1,ABF,CDF)
line_addition(1,BD,DF)
similar_triangle_property_line_ratio(1,ABF,CDF)
similar_triangle_property_line_ratio(1,FAB,FCD)
Problem: As shown in the diagram, AB=DiameterOfCircle(O), ∠AOC=50°, O is the center of circle O. Find the measure of ∠BCO.
问题：如图所示，AB=DiameterOfCircle(O)，∠AOC=50°，O是圆O的圆心。求∠BCO的大小。
Construction (CDL):
Shape(CO,OB,BC)
Shape(AO,OC,OCA)
Shape(DO,OA,OAD)
Shape(BO,OD,ODB)
Shape(CB,OBC)
Collinear(COD)
Collinear(AOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(AB),DiameterOfCircle(O))
Equal(MeasureOfAngle(AOC),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 25
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,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)
Problem: As shown in the diagram, ∠ACD=120°, BD∥EG. Find the measure of ∠EFC.
问题：如图所示，∠ACD=120°，BD∥EG。求∠EFC的大小。
Construction (CDL):
Shape(BC,CA)
Shape(AC,CD)
Shape(FC,CB)
Shape(DC,CF)
Shape(EF,FC)
Shape(CF,FG)
Shape(HF,FE)
Shape(GF,FH)
Collinear(BCD)
Collinear(EFG)
Collinear(ACFH)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),120)
ParallelBetweenLine(BD,EG)
Goal: Value(MeasureOfAngle(EFC))
Answer: 60
Theorem sequence:
parallel_property_collinear_extend(3,BD,EG,C)
parallel_property_collinear_extend(3,GE,DC,F)
parallel_property_corresponding_angle(1,CD,FG,A)
adjacent_complementary_angle(1,EFC,CFG)
Problem: As shown in the diagram, ∠ACD=62°, BD∥EG. Find the measure of ∠CFG.
问题：如图所示，∠ACD=62°，BD∥EG。求∠CFG的大小。
Construction (CDL):
Shape(BC,CA)
Shape(AC,CD)
Shape(FC,CB)
Shape(DC,CF)
Shape(EF,FC)
Shape(CF,FG)
Shape(HF,FE)
Shape(GF,FH)
Collinear(BCD)
Collinear(EFG)
Collinear(ACFH)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),62)
ParallelBetweenLine(BD,EG)
Goal: Value(MeasureOfAngle(CFG))
Answer: 62
Theorem sequence:
parallel_property_collinear_extend(3,BD,EG,C)
parallel_property_collinear_extend(3,GE,DC,F)
parallel_property_corresponding_angle(1,CD,FG,A)
Problem: As shown in the diagram, ∠BCE=20°, ∠GBA=46°, AB is parallel to GF, GF is parallel to CD. Find the measure of ∠FEC.
问题：如图所示，∠BCE=20°，∠GBA=46°，AB平行于GF，GF平行于CD。求∠FEC的大小。
Construction (CDL):
Shape(GB,BA)
Shape(BG,GE,EF,FB)
Shape(GC,CE,EG)
Shape(EC,CD,DF,FE)
Collinear(BGC)
Collinear(GEF)
Collinear(BFD)
Conditions (CDL):
Equal(MeasureOfAngle(BCE),20)
Equal(MeasureOfAngle(GBA),46)
ParallelBetweenLine(AB,GF)
ParallelBetweenLine(GF,CD)
Goal: Value(MeasureOfAngle(FEC))
Answer: 154
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,GF)
adjacent_complementary_angle(1,BGE,EGC)
triangle_property_angle_sum(1,GCE)
adjacent_complementary_angle(1,FEC,CEG)
Problem: As shown in the diagram, AB=6, the perimeter of △OCD is 16, AD and BC are opposite sides of the parallelogram DABC. Find the sum of the length of line AC and the length of line DB.
问题：如图所示，AB=6，三角形OCD的周长为16，AD和BC是平行四边形DABC的一组对边。求直线AC的长度与直线DB的长度之和。
Construction (CDL):
Shape(DA,AO,OD)
Shape(OA,AB,BO)
Shape(OB,BC,CO)
Shape(DO,OC,CD)
Collinear(DOB)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(PerimeterOfTriangle(OCD),16)
Parallelogram(DABC)
Goal: Value(Add(LengthOfLine(AC),LengthOfLine(DB)))
Answer: 20
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
triangle_perimeter_formula(1,OCD)
line_addition(1,DO,OB)
line_addition(1,AO,OC)
parallelogram_property_diagonal_bisection(1,DABC,O)
parallelogram_property_diagonal_bisection(1,ABCD,O)
Problem: As shown in the diagram, ∠BCA=35°, the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=35°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(BD,DA,OAB)
Shape(DB,BC,CD)
Shape(CB,OBC)
Shape(AO,OD,DC,OCA)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 70
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠AOC=35°, CO is perpendicular to DO. Find the measure of ∠DOB.
问题：如图所示，∠AOC=35°，CO⊥DO。求∠DOB的大小。
Construction (CDL):
Shape(CO,OD)
Shape(AO,OC)
Shape(DO,OB)
Collinear(AOB)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),35)
PerpendicularBetweenLine(CO,DO)
Goal: Value(MeasureOfAngle(DOB))
Answer: 55
Theorem sequence:
angle_addition(1,AOC,COD)
adjacent_complementary_angle(1,AOD,DOB)
Problem: As shown in the diagram, ∠OBD=25°, O is the center of circle O, BO∥CA. Find the measure of ∠DOB.
问题：如图所示，∠OBD=25°，O是圆O的圆心，BO平行于CA。求∠DOB的大小。
Construction (CDL):
Shape(OB,BD,DO)
Shape(CD,DB,OBC)
Shape(DC,CA,AD)
Shape(AC,OCA)
Shape(BO,OD,DA,OAB)
Collinear(ODC)
Collinear(ADB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBD),25)
IsCentreOfCircle(O,O)
ParallelBetweenLine(BO,CA)
Goal: Value(MeasureOfAngle(DOB))
Answer: 50
Theorem sequence:
parallel_property_alternate_interior_angle(1,BO,CA)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠AOC=65°. Find the measure of ∠COB.
问题：如图所示，∠AOC=65°。求∠COB的大小。
Construction (CDL):
Shape(AO,OC)
Shape(CO,OB)
Collinear(AOB)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),65)
Goal: Value(MeasureOfAngle(COB))
Answer: 115
Theorem sequence:
adjacent_complementary_angle(1,AOC,COB)
Problem: As shown in the diagram, ∠FAB=70°, CD∥AB. Find the measure of ∠CFE.
问题：如图所示，∠FAB=70°，CD∥AB。求∠CFE的大小。
Construction (CDL):
Shape(CF,FE)
Shape(EF,FD)
Shape(DF,FA)
Shape(AF,FC)
Shape(FA,AB)
Collinear(CFD)
Collinear(EFA)
Conditions (CDL):
Equal(MeasureOfAngle(FAB),70)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(CFE))
Answer: 110
Theorem sequence:
parallel_property_collinear_extend(3,CD,AB,F)
parallel_property_corresponding_angle(1,FD,AB,E)
adjacent_complementary_angle(1,CFE,EFD)
Problem: As shown in the diagram, ∠ABD=∠CDB, ∠DAB=60°. Find the measure of ∠CDA.
问题：如图所示，∠ABD=∠CDB，∠DAB=60°。求∠CDA的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(BC,CD,DB)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),MeasureOfAngle(CDB))
Equal(MeasureOfAngle(DAB),60)
Goal: Value(MeasureOfAngle(CDA))
Answer: 120
Theorem sequence:
parallel_judgment_alternate_interior_angle(1,BA,CD)
parallel_property_ipsilateral_internal_angle(1,DC,AB)
Problem: As shown in the diagram, ∠BEA=46°, ∠EAB=90°, BE is parallel to CF. Find the measure of ∠FCD.
问题：如图所示，∠BEA=46°，∠EAB=90°，BE平行于CF。求∠FCD的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(BC,CF,FE,EB)
Shape(CD,DG,GF,FC)
Collinear(ABCD)
Collinear(AEFG)
Conditions (CDL):
Equal(MeasureOfAngle(BEA),46)
Equal(MeasureOfAngle(EAB),90)
ParallelBetweenLine(BE,CF)
Goal: Value(MeasureOfAngle(FCD))
Answer: 136
Theorem sequence:
triangle_property_angle_sum(1,ABE)
parallel_property_corresponding_angle(1,BE,CF,A)
adjacent_complementary_angle(1,BCF,FCD)
Problem: As shown in the diagram, ∠NEB=36°, AB∥CD, GF is perpendicular to EF. Find the measure of ∠CFG.
问题：如图所示，∠NEB=36°，AB∥CD，GF⊥EF。求∠CFG的大小。
Construction (CDL):
Shape(AG,GM)
Shape(MG,GE)
Shape(GE,EN)
Shape(NE,EB)
Shape(FG,GA)
Shape(BE,EF)
Shape(CF,FG)
Shape(EF,FD)
Shape(PF,FC)
Shape(QF,FP)
Shape(DF,FQ)
Shape(GF,FE,EG)
Collinear(AGEB)
Collinear(CFD)
Collinear(MGFQ)
Collinear(PFEN)
Conditions (CDL):
Equal(MeasureOfAngle(NEB),36)
ParallelBetweenLine(AB,CD)
PerpendicularBetweenLine(GF,EF)
Goal: Value(MeasureOfAngle(CFG))
Answer: 54
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,N)
angle_addition(1,GFE,EFD)
adjacent_complementary_angle(1,CFG,GFD)
Problem: As shown in the diagram, AD=CD, ∠CDA=40°, AB is parallel to CD. Find the measure of ∠EAB.
问题：如图所示，AD=CD，∠CDA=40°，AB平行于CD。求∠EAB的大小。
Construction (CDL):
Shape(EA,AB)
Shape(AC,CD,DA)
Shape(BA,AD)
Collinear(EAC)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(CD))
Equal(MeasureOfAngle(CDA),40)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(EAB))
Answer: 70
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,DAC)
isosceles_triangle_property_angle_equal(1,DAC)
triangle_property_angle_sum(1,ACD)
parallel_property_corresponding_angle(1,AB,CD,E)
Problem: As shown in the diagram, △ABC is congruent to △HDC, ∠FAE=27°, ∠GBC=40°. Find the measure of ∠DCA.
问题：如图所示，三角形ABC与三角形HDC是全等三角形，∠FAE=27°，∠GBC=40°。求∠DCA的大小。
Construction (CDL):
Shape(AE,EF,FA)
Shape(DG,GE,ED)
Shape(BC,CG,GB)
Shape(HF,FC,CH)
Shape(EG,GC,CF,FE)
Collinear(AEGB)
Collinear(AFC)
Collinear(DEFH)
Collinear(DGC)
Collinear(BCH)
Conditions (CDL):
CongruentBetweenTriangle(ABC,HDC)
Equal(MeasureOfAngle(FAE),27)
Equal(MeasureOfAngle(GBC),40)
Goal: Value(MeasureOfAngle(DCA))
Answer: 46
Theorem sequence:
triangle_property_angle_sum(1,ABC)
adjacent_complementary_angle(1,BCA,ACH)
congruent_triangle_property_angle_equal(1,CAB,CHD)
angle_addition(1,DCF,FCH)
Problem: As shown in the diagram, ∠MEG=70°, NG bisects ∠END, AB is parallel to CD. Find the measure of ∠NGE.
问题：如图所示，∠MEG=70°，NG平分∠END，AB平行于CD。求∠NGE的大小。
Construction (CDL):
Shape(AE,EM)
Shape(ME,EB)
Shape(NE,EA)
Shape(GN,ND)
Shape(EN,NG,GE)
Shape(CN,NE)
Collinear(AEGB)
Collinear(MEN)
Collinear(CND)
Conditions (CDL):
Equal(MeasureOfAngle(MEG),70)
IsBisectorOfAngle(NG,END)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(NGE))
Answer: 35
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,E)
parallel_property_collinear_extend(3,DC,BE,N)
parallel_property_corresponding_angle(1,EB,ND,M)
angle_addition(1,ENG,GND)
adjacent_complementary_angle(1,MEG,GEN)
triangle_property_angle_sum(1,ENG)
Problem: As shown in the diagram, ∠CEA=90°, ∠ECD=44°, AB∥FE, FE is parallel to CD. Find the measure of ∠MAB.
问题：如图所示，∠CEA=90°，∠ECD=44°，AB∥FE，FE平行于CD。求∠MAB的大小。
Construction (CDL):
Shape(MA,AB)
Shape(BA,AE)
Shape(FE,EA)
Shape(CE,EF)
Shape(EC,CD)
Collinear(MAE)
Conditions (CDL):
Equal(MeasureOfAngle(CEA),90)
Equal(MeasureOfAngle(ECD),44)
ParallelBetweenLine(AB,FE)
ParallelBetweenLine(FE,CD)
Goal: Value(MeasureOfAngle(MAB))
Answer: 134
Theorem sequence:
parallel_property_alternate_interior_angle(2,FE,CD)
parallel_property_alternate_interior_angle(1,AB,FE)
angle_addition(1,CEF,FEA)
adjacent_complementary_angle(1,MAB,BAE)
Problem: As shown in the diagram, ∠COB=100°, O is the center of circle O. Find the measure of ∠BAC.
问题：如图所示，∠COB=100°，O是圆O的圆心。求∠BAC的大小。
Construction (CDL):
Shape(OB,BA,AC,CO)
Shape(AB,OBA)
Shape(CA,OAC)
Shape(BO,OC,OCB)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OBC,O)
round_arc(1,OBC,OCB)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠AHG=125°, AB∥CD, FE⊥CE. Find the measure of ∠FEI.
问题：如图所示，∠AHG=125°，AB∥CD，FE垂直于CE。求∠FEI的大小。
Construction (CDL):
Shape(AH,HG)
Shape(GH,HB)
Shape(EH,HA)
Shape(BH,HE)
Shape(CE,EH)
Shape(HE,ED)
Shape(IE,EC)
Shape(FE,EI)
Shape(DE,EF)
Collinear(AHB)
Collinear(CED)
Collinear(GHEI)
Conditions (CDL):
Equal(MeasureOfAngle(AHG),125)
ParallelBetweenLine(AB,CD)
PerpendicularBetweenLine(FE,CE)
Goal: Value(MeasureOfAngle(FEI))
Answer: 35
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,H)
parallel_property_collinear_extend(3,DC,HA,E)
parallel_property_corresponding_angle(2,EC,HA,G)
adjacent_complementary_angle(1,IEC,CEH)
angle_addition(1,FEI,IEC)
Problem: As shown in the diagram, ∠BAC=120°, AB is parallel to CD, CA is parallel to FD, CD∥EF. Find the measure of ∠FDC.
问题：如图所示，∠BAC=120°，AB平行于CD，CA平行于FD，CD平行于EF。求∠FDC的大小。
Construction (CDL):
Shape(BA,AC)
Shape(AC,CD)
Shape(FD,DC)
Shape(EF,FD)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),120)
ParallelBetweenLine(AB,CD)
ParallelBetweenLine(CA,FD)
ParallelBetweenLine(CD,EF)
Goal: Value(MeasureOfAngle(FDC))
Answer: 60
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AB,CD)
parallel_property_alternate_interior_angle(1,CA,FD)
Problem: As shown in the diagram, ∠CBA=70°, the center of circle D is D. Find the measure of ∠CDA.
问题：如图所示，∠CBA=70°，⊙D的圆心为D。求∠CDA的大小。
Construction (CDL):
Shape(BA,AD,DC,CB)
Shape(AB,OBA)
Shape(CD,DA,OAC)
Shape(BC,OCB)
Cocircular(D,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),70)
IsCentreOfCircle(D,D)
Goal: Value(MeasureOfAngle(CDA))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_external(1,DAC,B)
arc_property_center_angle(1,DAC,D)
Problem: As shown in the diagram, ∠DAC=35°, ∠DEA=115°. Find the measure of ∠ABC.
问题：如图所示，∠DAC=35°，∠DEA=115°。求∠ABC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(ED,ODE)
Shape(AE,OEA)
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Shape(AD,DE,EA)
Cocircular(O,ABCDE)
Conditions (CDL):
Equal(MeasureOfAngle(DAC),35)
Equal(MeasureOfAngle(DEA),115)
Goal: Value(MeasureOfAngle(ABC))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_internal(1,ODA,E)
arc_property_circumference_angle_external(1,ODA,C)
triangle_property_angle_sum(1,ACD)
arc_property_circumference_angle_internal(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠DAF=42°, EF is parallel to BG, CA is perpendicular to BA. Find the measure of ∠BCA.
问题：如图所示，∠DAF=42°，EF∥BG，CA⊥BA。求∠BCA的大小。
Construction (CDL):
Shape(DA,AF)
Shape(EA,AD)
Shape(FA,AC)
Shape(BA,AE)
Shape(AC,CG)
Shape(AB,BC,CA)
Collinear(EAF)
Collinear(BAD)
Collinear(BCG)
Conditions (CDL):
Equal(MeasureOfAngle(DAF),42)
ParallelBetweenLine(EF,BG)
PerpendicularBetweenLine(CA,BA)
Goal: Value(MeasureOfAngle(BCA))
Answer: 48
Theorem sequence:
parallel_property_collinear_extend(3,EF,BG,A)
parallel_property_corresponding_angle(1,AF,BG,D)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, triangle ABC is congruent to triangle DEF, AB=8, BE=4, DH=3, BA∥EH, BE⊥HE, ∠ABC is the right angle of the right △ ABC, ∠DEF is the right angle of the right △ DEF. Find the area of ABEH.
问题：如图所示，△ABC与△DEF是全等三角形，AB=8，BE=4，DH=3，BA∥EH，BE垂直于HE，△ABC为直角△，∠DEF是直角三角形DEF的直角。求ABEH的面积。
Construction (CDL):
Shape(AB,BE,EH,HA)
Shape(HE,EC,CH)
Shape(DH,HC,CF,FD)
Collinear(AHC)
Collinear(DHE)
Collinear(BECF)
Conditions (CDL):
CongruentBetweenTriangle(ABC,DEF)
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(BE),4)
Equal(LengthOfLine(DH),3)
ParallelBetweenLine(BA,EH)
PerpendicularBetweenLine(BE,HE)
RightTriangle(ABC)
RightTriangle(DEF)
Goal: Value(AreaOfQuadrilateral(ABEH))
Answer: 26
Theorem sequence:
congruent_triangle_property_line_equal(1,CAB,FDE)
line_addition(1,DH,HE)
trapezoid_judgment_parallel(1,BEHA)
right_trapezoid_judgment_right_angle(1,BEHA)
right_trapezoid_area_formular(1,BEHA)
Problem: As shown in the diagram, DB=10, DE=8, AD is the angle bisector of ∠CAB, AE is perpendicular to DE, DC is perpendicular to AC. Find the length of line BC.
问题：如图所示，DB=10，DE=8，AD平分∠CAB，AE垂直于DE，DC⊥AC。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(DB),10)
Equal(LengthOfLine(DE),8)
IsBisectorOfAngle(AD,CAB)
PerpendicularBetweenLine(AE,DE)
PerpendicularBetweenLine(DC,AC)
Goal: Value(LengthOfLine(BC))
Answer: 18
Theorem sequence:
mirror_congruent_triangle_judgment_aas(3,DCA,DAE)
mirror_congruent_triangle_property_line_equal(1,ADC,AED)
line_addition(1,CD,DB)
Problem: As shown in the diagram, AD=AC, ∠ABD=40°, ∠DAB=30°, DA is parallel to CE. Find the measure of ∠ACE.
问题：如图所示，AD=AC，∠ABD=40°，∠DAB=30°，DA∥CE。求∠ACE的大小。
Construction (CDL):
Shape(BD,DA,AB)
Shape(DC,CA,AD)
Shape(AC,CE)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(AC))
Equal(MeasureOfAngle(ABD),40)
Equal(MeasureOfAngle(DAB),30)
ParallelBetweenLine(DA,CE)
Goal: Value(MeasureOfAngle(ACE))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,BDA)
adjacent_complementary_angle(1,BDA,ADC)
isosceles_triangle_judgment_line_equal(1,ADC)
isosceles_triangle_property_angle_equal(1,ADC)
triangle_property_angle_sum(1,ADC)
parallel_property_alternate_interior_angle(2,DA,CE)
Problem: As shown in the diagram, ∠ABD=44°, ∠DCE=54°, AD bisects ∠CAB, ED is parallel to AB. Find the measure of ∠ADE.
问题：如图所示，∠ABD=44°，∠DCE=54°，AD平分∠CAB，ED平行于AB。求∠ADE的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Collinear(BDC)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),44)
Equal(MeasureOfAngle(DCE),54)
IsBisectorOfAngle(AD,CAB)
ParallelBetweenLine(ED,AB)
Goal: Value(MeasureOfAngle(ADE))
Answer: 41
Theorem sequence:
triangle_property_angle_sum(1,ABC)
angle_addition(1,EAD,DAB)
parallel_property_alternate_interior_angle(2,BA,DE)
Problem: As shown in the diagram, ∠BOC=140°. Find the measure of ∠COA.
问题：如图所示，∠BOC=140°。求∠COA的大小。
Construction (CDL):
Shape(CO,OA)
Shape(AO,OD)
Shape(DO,OB)
Shape(BO,OC)
Collinear(AOB)
Collinear(COD)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),140)
Goal: Value(MeasureOfAngle(COA))
Answer: 40
Theorem sequence:
adjacent_complementary_angle(1,BOC,COA)
Problem: As shown in the diagram, ∠ABE=50°, ∠ECA=90°, AE is the angle bisector of ∠CAB, DF∥AE. Find the measure of ∠FCE.
问题：如图所示，∠ABE=50°，∠ECA=90°，AE是∠CAB的角平分线，DF∥AE。求∠FCE的大小。
Construction (CDL):
Shape(AC,CD)
Shape(FC,CE)
Shape(CA,AE,EC)
Shape(AB,BE,EA)
Collinear(DCF)
Collinear(CEB)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),50)
Equal(MeasureOfAngle(ECA),90)
IsBisectorOfAngle(AE,CAB)
ParallelBetweenLine(DF,AE)
Goal: Value(MeasureOfAngle(FCE))
Answer: 70
Theorem sequence:
parallel_property_collinear_extend(3,DF,AE,C)
triangle_property_angle_sum(1,CAB)
angle_addition(1,CAE,EAB)
parallel_property_alternate_interior_angle(2,DC,AE)
angle_addition(1,ECA,ACD)
adjacent_complementary_angle(1,FCE,ECD)
Problem: As shown in the diagram, ∠AOE=90°, ∠DOB=50°. Find the measure of ∠COE.
问题：如图所示，∠AOE=90°，∠DOB=50°。求∠COE的大小。
Construction (CDL):
Shape(AO,OE)
Shape(EO,OD)
Shape(DO,OB)
Shape(BO,OC)
Shape(CO,OA)
Collinear(COD)
Collinear(AOB)
Conditions (CDL):
Equal(MeasureOfAngle(AOE),90)
Equal(MeasureOfAngle(DOB),50)
Goal: Value(MeasureOfAngle(COE))
Answer: 140
Theorem sequence:
vertical_angle(1,CPA,DOB)
angle_addition(1,COA,AOE)
Problem: As shown in the diagram, ∠BAE=45°, ∠DCF=90°, ∠EBA=90°, ∠EHG=30°, ∠FDC=60°, ∠GIF=45°. Find the measure of ∠IGH.
问题：如图所示，∠BAE=45°，∠DCF=90°，∠EBA=90°，∠EHG=30°，∠FDC=60°，∠GIF=45°。求∠IGH的大小。
Construction (CDL):
Shape(AE,EB,BA)
Shape(BE,EG,GF,FC)
Shape(CF,FD,DC)
Shape(EH,HG,GE)
Shape(GI,IF,FG)
Shape(IG,GH)
Collinear(ABCD)
Collinear(AEGI)
Collinear(DFGH)
Collinear(CFI)
Collinear(BEH)
Conditions (CDL):
Equal(MeasureOfAngle(BAE),45)
Equal(MeasureOfAngle(DCF),90)
Equal(MeasureOfAngle(EBA),90)
Equal(MeasureOfAngle(EHG),30)
Equal(MeasureOfAngle(FDC),60)
Equal(MeasureOfAngle(GIF),45)
Goal: Value(MeasureOfAngle(IGH))
Answer: 75
Theorem sequence:
triangle_property_angle_sum(1,CFD)
vertical_angle(1,CFD,IFG)
triangle_property_angle_sum(1,FGI)
adjacent_complementary_angle(1,FGI,IGH)
Problem: As shown in the diagram, ∠ACD=110°, ∠FCE=40°, BD∥EG. Find the measure of ∠CEF.
问题：如图所示，∠ACD=110°，∠FCE=40°，BD∥EG。求∠CEF的大小。
Construction (CDL):
Shape(BC,CA)
Shape(AC,CD)
Shape(EC,CB)
Shape(DC,CF)
Shape(CF,FG)
Shape(CE,EF,FC)
Collinear(ACF)
Collinear(BCD)
Collinear(EFG)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),110)
Equal(MeasureOfAngle(FCE),40)
ParallelBetweenLine(BD,EG)
Goal: Value(MeasureOfAngle(CEF))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,ACD,DCF)
parallel_property_collinear_extend(3,BD,EG,C)
parallel_property_collinear_extend(3,GE,DC,F)
parallel_property_ipsilateral_internal_angle(1,CD,FG)
adjacent_complementary_angle(1,EFC,CFG)
triangle_property_angle_sum(1,CEF)
Problem: As shown in the diagram, triangle ABC is congruent to triangle DEF, ∠ABE=45°, ∠CFD=65°. Find the measure of ∠COE.
问题：如图所示，△ABC与△DEF是全等三角形，∠ABE=45°，∠CFD=65°。求∠COE的大小。
Construction (CDL):
Shape(AB,BE,EO,OA)
Shape(DO,OC,CF,FD)
Shape(OE,EC,CO)
Collinear(AOC)
Collinear(DOE)
Collinear(BECF)
Conditions (CDL):
CongruentBetweenTriangle(ABC,DEF)
Equal(MeasureOfAngle(ABE),45)
Equal(MeasureOfAngle(CFD),65)
Goal: Value(MeasureOfAngle(COE))
Answer: 70
Theorem sequence:
congruent_triangle_property_angle_equal(1,BCA,EFD)
congruent_triangle_property_angle_equal(1,CAB,FDE)
triangle_property_angle_sum(1,OEC)
Problem: As shown in the diagram, AC=5, BC=3, ∠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，AC=5，BC=3，∠ABC是直角△ABC的直角。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(AC),5)
Equal(LengthOfLine(BC),3)
RightTriangle(ABC)
Goal: Value(Add(LengthOfLine(AB),LengthOfLine(BC)))
Answer: 7
Theorem sequence:
right_triangle_property_pythagorean(1,ABC)
Problem: As shown in the diagram, ∠CEF=45°, ∠ECD=30°, quadrilateral AHDB is a rectangle, ∠EFC is the right angle of the right triangle EFC. Find the measure of ∠GEF.
问题：如图所示，∠CEF=45°，∠ECD=30°，AHDB是长方形，△EFC为直角△。求∠GEF的大小。
Construction (CDL):
Shape(AH,HC,CG,GA)
Shape(FG,GE,EF)
Shape(EG,GC,CE)
Shape(BE,EC,CD,DB)
Collinear(AGEB)
Collinear(FGC)
Collinear(HCD)
Conditions (CDL):
Equal(MeasureOfAngle(CEF),45)
Equal(MeasureOfAngle(ECD),30)
Rectangle(AHDB)
RightTriangle(EFC)
Goal: Value(MeasureOfAngle(GEF))
Answer: 15
Theorem sequence:
parallel_property_collinear_extend(3,AB,HD,E)
parallel_property_collinear_extend(3,DH,EA,C)
parallel_property_alternate_interior_angle(2,DC,EA)
angle_addition(1,CEG,GEF)
Problem: As shown in the diagram, ∠NEM=37°, ∠PMA=56°, AB∥CD. Find the measure of ∠PQN.
问题：如图所示，∠NEM=37°，∠PMA=56°，AB平行于CD。求∠PQN的大小。
Construction (CDL):
Shape(EM,MN,NE)
Shape(MP,PQ,QN,NM)
Shape(AM,ME)
Shape(EN,NB)
Shape(PM,MA)
Shape(BN,NQ)
Shape(CP,PM)
Shape(FP,PC)
Shape(QP,PF)
Shape(GQ,QP)
Shape(DQ,QG)
Shape(NQ,QD)
Collinear(EMPF)
Collinear(ENQG)
Collinear(AMNB)
Collinear(CPQD)
Conditions (CDL):
Equal(MeasureOfAngle(NEM),37)
Equal(MeasureOfAngle(PMA),56)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(PQN))
Answer: 87
Theorem sequence:
parallel_property_collinear_extend(3,DC,BA,P)
parallel_property_collinear_extend(1,PC,BA,Q)
parallel_property_collinear_extend(3,AB,PQ,M)
parallel_property_alternate_interior_angle(2,AM,PQ)
triangle_property_angle_sum(1,EPQ)
Problem: As shown in the diagram, ∠MBA=55°, PQ∥MN, AB is perpendicular to CB. Find the measure of ∠FCQ.
问题：如图所示，∠MBA=55°，PQ∥MN，AB垂直于CB。求∠FCQ的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(PA,AE)
Shape(EA,AC)
Shape(AC,CF)
Shape(FC,CQ)
Shape(BA,AP)
Shape(QC,CB)
Shape(MB,BA)
Shape(CB,BN)
Collinear(PACQ)
Collinear(MBN)
Collinear(EAB)
Collinear(FCB)
Conditions (CDL):
Equal(MeasureOfAngle(MBA),55)
ParallelBetweenLine(PQ,MN)
PerpendicularBetweenLine(AB,CB)
Goal: Value(MeasureOfAngle(FCQ))
Answer: 35
Theorem sequence:
parallel_property_collinear_extend(3,PQ,MN,C)
parallel_property_collinear_extend(1,CQ,MN,A)
parallel_property_collinear_extend(3,NM,CA,B)
parallel_property_alternate_interior_angle(1,AC,MB)
triangle_property_angle_sum(1,ABC)
vertical_angle(1,FCQ,BCP)