SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠LWX=53°, WL is parallel to XE, XN∥ZK. Find the measure of ∠XZK.
问题：如图所示，∠LWX=53°，WL平行于XE，XN平行于ZK。求∠XZK的大小。
Construction (CDL):
Shape(GW,WL)
Shape(LW,WX)
Shape(WX,XE)
Shape(EX,XN)
Shape(NX,XZ)
Shape(XZ,ZK)
Shape(KZ,ZH)
Shape(HZ,ZY)
Shape(ZY,YM)
Shape(MY,YI)
Shape(IY,YW)
Shape(YW,WG)
Shape(WY,YZ,ZX,XW)
Collinear(GWXN)
Collinear(IYZK)
Collinear(LWYM)
Collinear(EXZH)
Conditions (CDL):
Equal(MeasureOfAngle(LWX),53)
ParallelBetweenLine(WL,XE)
ParallelBetweenLine(XN,ZK)
Goal: Value(MeasureOfAngle(XZK))
Answer: 53
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,WL,XE)
vertical_angle(1,WXE,NXZ)
parallel_property_ipsilateral_internal_angle(1,XN,ZK)
Problem: As shown in the diagram, ∠MRQ=5*x°, ∠PMR=x°, ∠QPM=4*x°, ∠RQP=2*x°. Find the measure of ∠PMR.
问题：如图所示，∠MRQ=5*x°，∠PMR=x°，∠QPM=4*x°，∠RQP=2*x°。求∠PMR的大小。
Construction (CDL):
Shape(MR,RQ,QP,PM)
Conditions (CDL):
Equal(MeasureOfAngle(MRQ),5*x)
Equal(MeasureOfAngle(PMR),x)
Equal(MeasureOfAngle(QPM),4*x)
Equal(MeasureOfAngle(RQP),2*x)
Goal: Value(MeasureOfAngle(PMR))
Answer: 30
Theorem sequence:
quadrilateral_property_angle_sum(1,MRQP)
Problem: As shown in the diagram, AB=15, BC=h, ∠CAB=45°, BC is perpendicular to AC. Find the value of h.
问题：如图所示，AB=15，BC=h，∠CAB=45°，BC垂直于AC。求h的值。
Construction (CDL):
Shape(CA,AB,BC)
Conditions (CDL):
Equal(LengthOfLine(AB),15)
Equal(LengthOfLine(BC),h)
Equal(MeasureOfAngle(CAB),45)
PerpendicularBetweenLine(BC,AC)
Goal: Value(h)
Answer: 15*sqrt(2)/2
Theorem sequence:
sine_theorem(1,BCA)
Problem: As shown in the diagram, AC=18, AD=24, BC=x, BD=9, ∠BAC=∠DAB. Find the value of x.
问题：如图所示，AC=18，AD=24，BC=x，BD=9，∠BAC=∠DAB。求x的值。
Construction (CDL):
Shape(CB,BA,AC)
Shape(BD,DA,AB)
Collinear(CBD)
Conditions (CDL):
Equal(LengthOfLine(AC),18)
Equal(LengthOfLine(AD),24)
Equal(LengthOfLine(BC),x)
Equal(LengthOfLine(BD),9)
Equal(MeasureOfAngle(BAC),MeasureOfAngle(DAB))
Goal: Value(x)
Answer: 27/4
Theorem sequence:
bisector_of_angle_judgment_angle_equal(1,AB,DAC)
bisector_of_angle_property_line_ratio(1,AB,DAC)
Problem: As shown in the diagram, AB=2*x+3, BC=5*x, ∠CBA=80°, quadrilateral ADCB is a rhombus. Find the measure of ∠DCB.
问题：如图所示，AB=2*x+3，BC=5*x，∠CBA=80°，ADCB是菱形。求∠DCB的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Shape(EC,CB,BE)
Shape(AE,EB,BA)
Collinear(AEC)
Collinear(DEB)
Conditions (CDL):
Equal(LengthOfLine(AB),2*x+3)
Equal(LengthOfLine(BC),5*x)
Equal(MeasureOfAngle(CBA),80)
Rhombus(ADCB)
Goal: Value(MeasureOfAngle(DCB))
Answer: 100
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,CD,BA)
Problem: As shown in the diagram, ∠PSR=x+10°, ∠QPS=x°, ∠RQP=2*x-16°, ∠SRQ=2*x°. Find the measure of ∠SRQ.
问题：如图所示，∠PSR=x+10°，∠QPS=x°，∠RQP=2*x-16°，∠SRQ=2*x°。求∠SRQ的大小。
Construction (CDL):
Shape(QP,PS,SR,RQ)
Conditions (CDL):
Equal(MeasureOfAngle(PSR),x+10)
Equal(MeasureOfAngle(QPS),x)
Equal(MeasureOfAngle(RQP),2*x-16)
Equal(MeasureOfAngle(SRQ),2*x)
Goal: Value(MeasureOfAngle(SRQ))
Answer: 122
Theorem sequence:
quadrilateral_property_angle_sum(1,QPSR)
Problem: As shown in the diagram, KL=NL, NM=ML, ∠JLK=25°, ∠KLN=18°, ∠NKJ=130°, ∠NLM=20°, LN and LK are the legs of the isosceles triangle LNK, triangleMNL is an isosceles triangle. Find the measure of ∠LMN.
问题：如图所示，KL=NL，NM=ML，∠JLK=25°，∠KLN=18°，∠NKJ=130°，∠NLM=20°，LN和LK是等腰△LNK的腰，∠MNL和∠NLM是等腰三角形MNL的底角。求∠LMN的大小。
Construction (CDL):
Shape(JL,LK,KJ)
Shape(KL,LN,NK)
Shape(NL,LM,MN)
Conditions (CDL):
Equal(LengthOfLine(KL),LengthOfLine(NL))
Equal(LengthOfLine(NM),LengthOfLine(ML))
Equal(MeasureOfAngle(JLK),25)
Equal(MeasureOfAngle(KLN),18)
Equal(MeasureOfAngle(NKJ),130)
Equal(MeasureOfAngle(NLM),20)
IsoscelesTriangle(LNK)
IsoscelesTriangle(MNL)
Goal: Value(MeasureOfAngle(LMN))
Answer: 140
Theorem sequence:
isosceles_triangle_property_angle_equal(1,MNL)
triangle_property_angle_sum(1,NLM)
Problem: As shown in the diagram, ∠AGC=60°, DG is perpendicular to AG. Find the measure of ∠EGD.
问题：如图所示，∠AGC=60°，DG⊥AG。求∠EGD的大小。
Construction (CDL):
Shape(GC,GCA,AG)
Shape(GA,GAD,DG)
Shape(GD,GDE,EG)
Shape(GE,GEB,BG)
Shape(GB,GBC,CG)
Collinear(AGB)
Collinear(CGE)
Cocircular(G,CADEB)
Conditions (CDL):
Equal(MeasureOfAngle(AGC),60)
PerpendicularBetweenLine(DG,AG)
Goal: Value(MeasureOfAngle(EGD))
Answer: 30
Theorem sequence:
adjacent_complementary_angle(1,AGC,CGB)
adjacent_complementary_angle(1,BGD,DGA)
adjacent_complementary_angle(1,CGB,BGE)
angle_addition(1,BGE,EGD)
Problem: As shown in the diagram, QR=2, QW=15, ST=5, XW=12, WS∥VT, XR∥WS. Find the length of line WV.
问题：如图所示，QR=2，QW=15，ST=5，XW=12，WS平行于VT，XR平行于WS。求直线WV的长度。
Construction (CDL):
Shape(QX,XR,RQ)
Shape(XW,WS,SR,RX)
Shape(WV,VT,TS,SW)
Collinear(QXWV)
Collinear(QRST)
Conditions (CDL):
Equal(LengthOfLine(QR),2)
Equal(LengthOfLine(QW),15)
Equal(LengthOfLine(ST),5)
Equal(LengthOfLine(XW),12)
ParallelBetweenLine(WS,VT)
ParallelBetweenLine(XR,WS)
Goal: Value(LengthOfLine(WV))
Answer: 15/2
Theorem sequence:
parallel_property_corresponding_angle(1,XR,WS,Q)
parallel_property_corresponding_angle(1,WS,VT,X)
similar_triangle_judgment_aa(1,RQX,SQW)
line_addition(1,QX,XW)
similar_triangle_property_line_ratio(1,RQX,SQW)
similar_triangle_property_line_ratio(1,XRQ,WSQ)
similar_triangle_judgment_aa(1,SQW,TQV)
line_addition(1,QS,ST)
line_addition(1,QW,WV)
similar_triangle_property_line_ratio(1,SQW,TQV)
similar_triangle_property_line_ratio(1,WSQ,VTQ)
Problem: As shown in the diagram, ∠EAD=42°, A is the center of circle A. Find the measure of ⌒AEC.
问题：如图所示，∠EAD=42°，圆A的圆心为A。求弧AEC的角度。
Construction (CDL):
Shape(AE,AEB,BA)
Shape(AB,ABC,CA)
Shape(AC,ACD,DA)
Shape(AD,ADE,EA)
Collinear(BAD)
Collinear(CAE)
Cocircular(A,BCDE)
Conditions (CDL):
Equal(MeasureOfAngle(EAD),42)
IsCentreOfCircle(A,A)
Goal: Value(MeasureOfArc(AEC))
Answer: 180
Theorem sequence:
flat_angle(1,CAE)
arc_property_center_angle(1,AEC,A)
Problem: As shown in the diagram, CA=15, CB=x, ∠GBA=30°, the center of ⊙C is C, the tangent to circle C is BA. Find the value of x.
问题：如图所示，CA=15，CB=x，∠GBA=30°，C是圆C的圆心，BA是圆O的切线。求x的值。
Construction (CDL):
Shape(BA,CGA,GB)
Shape(AC,CG,CGA)
Shape(CA,CAG,GC)
Collinear(CGB)
Cocircular(C,AG)
Conditions (CDL):
Equal(LengthOfLine(CA),15)
Equal(LengthOfLine(CB),x)
Equal(MeasureOfAngle(GBA),30)
IsCentreOfCircle(C,C)
IsTangentOfCircle(BA,C)
Goal: Value(x)
Answer: 30
Theorem sequence:
tangent_of_circle_property_perpendicular(2,BA,C,C)
sine_theorem(1,CBA)
Problem: As shown in the diagram, ∠CAD=42°, ∠CFG=77°, △CDE is an equilateral △, triangleCAB is an isosceles triangle, △CFG is an isosceles △. Find the measure of ∠FCD.
问题：如图所示，∠CAD=42°，∠CFG=77°，△CDE为等边△，∠CAB和∠ABC是等腰三角形CAB的底角，∠CFG和∠FGC是等腰△CFG的底角。求∠FCD的大小。
Construction (CDL):
Shape(CA,AD,DC)
Shape(CD,DF,FC)
Shape(CF,FG,GC)
Shape(CG,GE,EC)
Shape(CE,EB,BC)
Collinear(ADFGEB)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),42)
Equal(MeasureOfAngle(CFG),77)
EquilateralTriangle(CDE)
IsoscelesTriangle(CAB)
IsoscelesTriangle(CFG)
Goal: Value(MeasureOfAngle(FCD))
Answer: 17
Theorem sequence:
angle_addition(1,DFC,CFG)
flat_angle(1,AFG)
equilateral_triangle_property_angle(1,DEC)
triangle_property_angle_sum(1,CDF)
Problem: As shown in the diagram, AB=y, AC=5, BC=x, ∠BAC=60°, AC⊥BC. Find the value of x.
问题：如图所示，AB=y，AC=5，BC=x，∠BAC=60°，AC垂直于BC。求x的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),y)
Equal(LengthOfLine(AC),5)
Equal(LengthOfLine(BC),x)
Equal(MeasureOfAngle(BAC),60)
PerpendicularBetweenLine(AC,BC)
Goal: Value(x)
Answer: 5*sqrt(3)
Theorem sequence:
triangle_property_angle_sum(1,ACB)
sine_theorem(1,ACB)
sine_theorem(1,BAC)
Problem: As shown in the diagram, CB=18, CD=12, ∠BAD=115°, AB and DC are opposite sides of the ▱ ADCB. Find the measure of ∠ADC.
问题：如图所示，CB=18，CD=12，∠BAD=115°，四边形ADCB是▱。求∠ADC的大小。
Construction (CDL):
Shape(AD,DC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(CB),18)
Equal(LengthOfLine(CD),12)
Equal(MeasureOfAngle(BAD),115)
Parallelogram(ADCB)
Goal: Value(MeasureOfAngle(ADC))
Answer: 65
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AB,DC)
Problem: As shown in the diagram, QR=22, XY=15, XY is the midsegment of QTSR, quadrilateral QTSR is a isosceles trapezoid. Find the length of line TS.
问题：如图所示，QR=22，XY=15，四边形QTSR的中位线为XY，QTSR是等腰梯形。求直线TS的长度。
Construction (CDL):
Shape(QX,XY,YR,RQ)
Shape(XT,TS,SY,YX)
Collinear(QXT)
Collinear(RYS)
Conditions (CDL):
Equal(LengthOfLine(QR),22)
Equal(LengthOfLine(XY),15)
IsMidsegmentOfQuadrilateral(XY,QTSR)
IsoscelesTrapezoid(QTSR)
Goal: Value(LengthOfLine(TS))
Answer: 8
Theorem sequence:
midsegment_of_quadrilateral_property_length(1,XY,QTSR)
Problem: As shown in the diagram, ∠FHM=94°, HM∥CA. Find the measure of ∠HCA.
问题：如图所示，∠FHM=94°，HM∥CA。求∠HCA的大小。
Construction (CDL):
Shape(FH,HM)
Shape(MH,HC)
Shape(HC,CA)
Shape(AC,CI)
Shape(IC,CE)
Shape(EC,CH)
Shape(CH,HD)
Shape(DH,HF)
Collinear(FHCI)
Collinear(MHD)
Collinear(ACE)
Conditions (CDL):
Equal(MeasureOfAngle(FHM),94)
ParallelBetweenLine(HM,CA)
Goal: Value(MeasureOfAngle(HCA))
Answer: 94
Theorem sequence:
parallel_property_corresponding_angle(1,HM,CA,F)
Problem: As shown in the diagram, ∠YZW=60°, ∠ZWX=95°. Find the measure of ∠XYZ.
问题：如图所示，∠YZW=60°，∠ZWX=95°。求∠XYZ的大小。
Construction (CDL):
Shape(VZW,WZ)
Shape(VWX,XW)
Shape(WXY,YX)
Shape(VYZ,ZY)
Shape(ZW,WX,XY,YZ)
Cocircular(V,ZWXY)
Conditions (CDL):
Equal(MeasureOfAngle(YZW),60)
Equal(MeasureOfAngle(ZWX),95)
Goal: Value(MeasureOfAngle(XYZ))
Answer: 85
Theorem sequence:
arc_property_circumference_angle_external(1,VZX,Y)
arc_property_circumference_angle_internal(1,VZX,W)
Problem: As shown in the diagram, ∠ADG=36°, ∠AGF=104°, ∠EFC=40°, GB⊥CB. Find the measure of ∠DGA.
问题：如图所示，∠ADG=36°，∠AGF=104°，∠EFC=40°，GB⊥CB。求∠DGA的大小。
Construction (CDL):
Shape(AD,DG,GA)
Shape(AG,GF)
Shape(GF,FE)
Shape(EF,FC)
Shape(BG,GD)
Shape(GB,BF,FG)
Shape(FB,BC,CF)
Collinear(DGFC)
Collinear(BFE)
Conditions (CDL):
Equal(MeasureOfAngle(ADG),36)
Equal(MeasureOfAngle(AGF),104)
Equal(MeasureOfAngle(EFC),40)
PerpendicularBetweenLine(GB,CB)
Goal: Value(MeasureOfAngle(DGA))
Answer: 76
Theorem sequence:
adjacent_complementary_angle(1,DGA,AGF)
Problem: As shown in the diagram, the area of XWZY is 100, XZ=10, XWZY is a rhombus. Find the length of line WY.
问题：如图所示，四边形XWZY的面积为100，XZ=10，XWZY是菱形。求直线WY的长度。
Construction (CDL):
Shape(XW,WA,AX)
Shape(AW,WZ,ZA)
Shape(AZ,ZY,YA)
Shape(XA,AY,YX)
Collinear(WAY)
Collinear(XAZ)
Conditions (CDL):
Equal(AreaOfQuadrilateral(XWZY),100)
Equal(LengthOfLine(XZ),10)
Rhombus(XWZY)
Goal: Value(LengthOfLine(WY))
Answer: 20
Theorem sequence:
kite_area_formula_diagonal(1,XWZY)
Problem: As shown in the diagram, CB∥DE. Prove that ∠TCB+∠EDG=180°.
问题：如图所示，CB∥DE。求证∠TCB+∠EDG=180°。
Construction (CDL):
Shape(TC,CB)
Shape(CD,DE,EB,BC)
Shape(ED,DG)
Collinear(TCDG)
Conditions (CDL):
ParallelBetweenLine(CB,DE)
Goal: Equal(Add(MeasureOfAngle(TCB),MeasureOfAngle(EDG)),180)
Answer: 0
Theorem sequence:
parallel_property_corresponding_angle(1,CB,DE,T)
adjacent_complementary_angle(1,CDE,EDG)
Problem: As shown in the diagram, AC=3, ∠DAC=∠BAE, ∠EAD=130°, the center of circle A is A. Find the sum of the area of the sector ACD and the area of the sector AEB.
问题：如图所示，AC=3，∠DAC=∠BAE，∠EAD=130°，A是⊙A的圆心。求扇形ACD的面积与扇形AEB的面积之和。
Construction (CDL):
Shape(ACD,DA,AC)
Shape(AD,ADE,EA)
Shape(AE,AEB,BA)
Shape(AB,ABC,CA)
Collinear(CAE)
Collinear(DAB)
Cocircular(A,CDEB)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
Equal(MeasureOfAngle(DAC),MeasureOfAngle(BAE))
Equal(MeasureOfAngle(EAD),130)
IsCentreOfCircle(A,A)
Goal: Value(Add(AreaOfSector(ACD),AreaOfSector(AEB)))
Answer: 5*pi/2
Theorem sequence:
adjacent_complementary_angle(1,BAE,EAD)
radius_of_circle_property_length_equal(1,AC,A)
arc_property_center_angle(1,ACD,A)
arc_property_center_angle(1,AEB,A)
sector_area_formula(1,ACD)
sector_area_formula(1,AEB)
Problem: As shown in the diagram, ∠SRU=6*x-54°, ∠UTS=4*x+6°, quadrilateral TSRU is a parallelogram. Find the value of x.
问题：如图所示，∠SRU=6*x-54°，∠UTS=4*x+6°，TSRU是平行四边形。求x的值。
Construction (CDL):
Shape(TS,SR,RU,UT)
Conditions (CDL):
Equal(MeasureOfAngle(SRU),6*x-54)
Equal(MeasureOfAngle(UTS),4*x+6)
Parallelogram(TSRU)
Goal: Value(x)
Answer: 30
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,TSRU)
Problem: As shown in the diagram, AC=27, BA=2*x+5, BA=BC, BC=3*x-4. Find the length of line BC.
问题：如图所示，AC=27，BA=2*x+5，BA=BC，BC=3*x-4。求直线BC的长度。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AC),27)
Equal(LengthOfLine(BA),2*x+5)
Equal(LengthOfLine(BA),LengthOfLine(BC))
Equal(LengthOfLine(BC),3*x-4)
Goal: Value(LengthOfLine(BC))
Answer: 23
Theorem sequence:
Problem: As shown in the diagram, Add(PerimeterOfCircle(A)=PerimeterOfCircle(B), the radius of circle A is Mul(RadiusOfCircle(B),2), the radius of ⊙A is Mul(RadiusOfCircle(C),4), A is the center of circle A, the center of ⊙B is B, the center of circle C is C. Find the length of line AC.
问题：如图所示，Add(PerimeterOfCircle(A)=PerimeterOfCircle(B)，圆A的半径为Mul(RadiusOfCircle(B),2)，圆A的半径为Mul(RadiusOfCircle(C),4)，⊙A的圆心为A，B是圆B的圆心，C是圆C的圆心。求直线AC的长度。
Construction (CDL):
Shape(BD,BDE,EB)
Shape(BE,BED,DB)
Collinear(CDBEA)
Cocircular(C,D)
Cocircular(B,DE)
Cocircular(A,E)
Conditions (CDL):
Equal(Add(PerimeterOfCircle(A),PerimeterOfCircle(B),PerimeterOfCircle(C)),42*pi)
Equal(RadiusOfCircle(A),Mul(RadiusOfCircle(B),2))
Equal(RadiusOfCircle(A),Mul(RadiusOfCircle(C),4))
IsCentreOfCircle(A,A)
IsCentreOfCircle(B,B)
IsCentreOfCircle(C,C)
Goal: Value(LengthOfLine(AC))
Answer: 27
Theorem sequence:
circle_perimeter_formula(1,A)
circle_perimeter_formula(1,B)
circle_perimeter_formula(1,C)
radius_of_circle_property_length_equal(1,CD,C)
radius_of_circle_property_length_equal(1,BD,B)
radius_of_circle_property_length_equal(1,BE,B)
radius_of_circle_property_length_equal(1,AE,A)
line_addition(1,CD,DB)
line_addition(1,CB,BE)
line_addition(1,CE,EA)
Problem: As shown in the diagram, AE=12, AJ=15, ∠JBE=34°, ∠JCD=32°, the incenter of △CBA is J, AF⊥JF, CD⊥JD, JE⊥AE. Find the measure of ∠JAC.
问题：如图所示，AE=12，AJ=15，∠JBE=34°，∠JCD=32°，△CBA的内心为J，AF垂直于JF，CD垂直于JD，JE垂直于AE。求∠JAC的大小。
Construction (CDL):
Shape(CD,DJ,JC)
Shape(DB,BJ,JD)
Shape(JB,BE,EJ)
Shape(JE,EA,AJ)
Shape(JA,AF,FJ)
Shape(JF,FC,CJ)
Collinear(CDB)
Collinear(BEA)
Collinear(AFC)
Conditions (CDL):
Equal(LengthOfLine(AE),12)
Equal(LengthOfLine(AJ),15)
Equal(MeasureOfAngle(JBE),34)
Equal(MeasureOfAngle(JCD),32)
IsIncenterOfTriangle(J,CBA)
PerpendicularBetweenLine(AF,JF)
PerpendicularBetweenLine(CD,JD)
PerpendicularBetweenLine(JE,AE)
Goal: Value(MeasureOfAngle(JAC))
Answer: 24
Theorem sequence:
angle_addition(1,FCJ,JCD)
angle_addition(1,DBJ,JBE)
angle_addition(1,EAJ,JAF)
triangle_property_angle_sum(1,CBA)
Problem: As shown in the diagram, ∠UVT=23°. Find the measure of ⌒DTU.
问题：如图所示，∠UVT=23°。求⌒DTU的角度。
Construction (CDL):
Shape(DVT,TV)
Shape(VT,DTU,UV)
Shape(VU,DUV)
Cocircular(D,VTU)
Conditions (CDL):
Equal(MeasureOfAngle(UVT),23)
Goal: Value(MeasureOfArc(DTU))
Answer: 46
Theorem sequence:
arc_property_circumference_angle_external(1,DTU,V)
Problem: As shown in the diagram, VW=3*x-6, WX=x+4, YW=5, ZW=6, YX∥VZ. Find the length of line WX.
问题：如图所示，VW=3*x-6，WX=x+4，YW=5，ZW=6，YX∥VZ。求直线WX的长度。
Construction (CDL):
Shape(VZ,ZW,WV)
Shape(WX,XY,YW)
Collinear(ZWY)
Collinear(VWX)
Conditions (CDL):
Equal(LengthOfLine(VW),3*x-6)
Equal(LengthOfLine(WX),x+4)
Equal(LengthOfLine(YW),5)
Equal(LengthOfLine(ZW),6)
ParallelBetweenLine(YX,VZ)
Goal: Value(LengthOfLine(WX))
Answer: 10
Theorem sequence:
parallel_property_alternate_interior_angle(1,YX,VZ)
parallel_property_alternate_interior_angle(2,YX,VZ)
similar_triangle_judgment_aa(1,WVZ,WXY)
similar_triangle_property_line_ratio(1,VZW,XYW)
similar_triangle_property_line_ratio(1,ZWV,YWX)
Problem: As shown in the diagram, ∠SRU=23°, the measure of arc BVT is 68. Find the measure of arc BSU.
问题：如图所示，∠SRU=23°，弧BVT的角度为68。求弧BSU的角度。
Construction (CDL):
Shape(SR,RU,BSU)
Shape(ST,BTS)
Shape(BSU,UV,BVT,TS)
Shape(BUV,VU)
Collinear(RST)
Collinear(RUV)
Cocircular(B,SUVT)
Conditions (CDL):
Equal(MeasureOfAngle(SRU),23)
Equal(MeasureOfArc(BVT),68)
Goal: Value(MeasureOfArc(BSU))
Answer: 22
Theorem sequence:
circle_property_circular_power_segment_and_segment_angle(1,RST,RUV,B)
Problem: As shown in the diagram, AB=12, AC=y, AD=4, BC=x, CD=z, BA⊥CA, DC⊥BC. Find the value of x.
问题：如图所示，AB=12，AC=y，AD=4，BC=x，CD=z，BA垂直于CA，DC⊥BC。求x的值。
Construction (CDL):
Shape(CB,BA,AC)
Shape(CA,AD,DC)
Collinear(BAD)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(AC),y)
Equal(LengthOfLine(AD),4)
Equal(LengthOfLine(BC),x)
Equal(LengthOfLine(CD),z)
PerpendicularBetweenLine(BA,CA)
PerpendicularBetweenLine(DC,BC)
Goal: Value(x)
Answer: 8*sqrt(3)
Theorem sequence:
line_addition(1,BA,AD)
mirror_similar_triangle_judgment_aa(1,CBA,DCB)
mirror_similar_triangle_property_line_ratio(1,ACB,CBD)
mirror_similar_triangle_property_line_ratio(1,CBA,DCB)
Problem: As shown in the diagram, AE=5, AF=x, AH=6, GA=12. Find the value of x.
问题：如图所示，AE=5，AF=x，AH=6，GA=12。求x的值。
Construction (CDL):
Shape(AG,JGF,FA)
Shape(AF,JFE,EA)
Shape(AE,JEH,HA)
Shape(AH,JHG,GA)
Collinear(FAH)
Collinear(GAE)
Cocircular(J,FEHG)
Conditions (CDL):
Equal(LengthOfLine(AE),5)
Equal(LengthOfLine(AF),x)
Equal(LengthOfLine(AH),6)
Equal(LengthOfLine(GA),12)
Goal: Value(x)
Answer: 10
Theorem sequence:
circle_property_circular_power_chord_and_chord(1,FAH,GAE,J)
Problem: As shown in the diagram, ∠LWX=53°, WL is parallel to XE, XN∥ZK. Find the measure of ∠ZYM.
问题：如图所示，∠LWX=53°，WL∥XE，XN∥ZK。求∠ZYM的大小。
Construction (CDL):
Shape(GW,WL)
Shape(LW,WX)
Shape(WX,XE)
Shape(EX,XN)
Shape(NX,XZ)
Shape(XZ,ZK)
Shape(KZ,ZH)
Shape(HZ,ZY)
Shape(ZY,YM)
Shape(MY,YI)
Shape(IY,YW)
Shape(YW,WG)
Shape(WY,YZ,ZX,XW)
Collinear(GWXN)
Collinear(IYZK)
Collinear(MYWL)
Collinear(HZXE)
Conditions (CDL):
Equal(MeasureOfAngle(LWX),53)
ParallelBetweenLine(WL,XE)
ParallelBetweenLine(XN,ZK)
Goal: Value(MeasureOfAngle(ZYM))
Answer: 127
Theorem sequence:
parallel_property_collinear_extend(1,XN,ZK,W)
parallel_property_collinear_extend(2,KZ,XW,Y)
adjacent_complementary_angle(1,LWX,XWY)
parallel_property_corresponding_angle(2,WX,YZ,M)
Problem: As shown in the diagram, AC=11, BC=x, ∠ABC=30°, ∠CAB=120°. Find the value of x.
问题：如图所示，AC=11，BC=x，∠ABC=30°，∠CAB=120°。求x的值。
Construction (CDL):
Shape(CA,AB,BC)
Conditions (CDL):
Equal(LengthOfLine(AC),11)
Equal(LengthOfLine(BC),x)
Equal(MeasureOfAngle(ABC),30)
Equal(MeasureOfAngle(CAB),120)
Goal: Value(x)
Answer: 11*sqrt(3)
Theorem sequence:
sine_theorem(1,CAB)
Problem: As shown in the diagram, ∠TWV=3*x-4°, ∠UTW=x°, ∠VUT=3*x-4°, ∠WVU=x°. Find the measure of ∠UTW.
问题：如图所示，∠TWV=3*x-4°，∠UTW=x°，∠VUT=3*x-4°，∠WVU=x°。求∠UTW的大小。
Construction (CDL):
Shape(UT,TW,WV,VU)
Conditions (CDL):
Equal(MeasureOfAngle(TWV),3*x-4)
Equal(MeasureOfAngle(UTW),x)
Equal(MeasureOfAngle(VUT),3*x-4)
Equal(MeasureOfAngle(WVU),x)
Goal: Value(MeasureOfAngle(UTW))
Answer: 46
Theorem sequence:
quadrilateral_property_angle_sum(1,UTWV)
Problem: As shown in the diagram, ∠ABD=x+14°, ∠ABF=3*x-8°, BD bisects ∠ABF. Find the measure of ∠ABD.
问题：如图所示，∠ABD=x+14°，∠ABF=3*x-8°，BD平分∠ABF。求∠ABD的大小。
Construction (CDL):
Shape(AB,BD)
Shape(DB,BF)
Shape(FB,BC)
Collinear(ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),x+14)
Equal(MeasureOfAngle(ABF),3*x-8)
IsBisectorOfAngle(BD,ABF)
Goal: Value(MeasureOfAngle(ABD))
Answer: 50
Theorem sequence:
angle_addition(1,ABD,DBF)
Problem: As shown in the diagram, AJ=x, AK=x+2, AM=x+7, LA=x+10. Find the value of x.
问题：如图所示，AJ=x，AK=x+2，AM=x+7，LA=x+10。求x的值。
Construction (CDL):
Shape(AJ,OJM,MA)
Shape(AM,OML,LA)
Shape(AL,OLK,KA)
Shape(AK,OKJ,JA)
Collinear(JAL)
Collinear(KAM)
Cocircular(O,MLKJ)
Conditions (CDL):
Equal(LengthOfLine(AJ),x)
Equal(LengthOfLine(AK),x+2)
Equal(LengthOfLine(AM),x+7)
Equal(LengthOfLine(LA),x+10)
Goal: Value(x)
Answer: 14
Theorem sequence:
circle_property_circular_power_chord_and_chord(1,JAL,KAM,O)
Problem: As shown in the diagram, CB=8, CF=x, FB=15, BC⊥FC. Find the value of x.
问题：如图所示，CB=8，CF=x，FB=15，BC垂直于FC。求x的值。
Construction (CDL):
Shape(BC,CF,FB)
Conditions (CDL):
Equal(LengthOfLine(CB),8)
Equal(LengthOfLine(CF),x)
Equal(LengthOfLine(FB),15)
PerpendicularBetweenLine(BC,FC)
Goal: Value(x)
Answer: sqrt(161)
Theorem sequence:
right_triangle_judgment_angle(1,BCF)
right_triangle_property_pythagorean(1,BCF)
Problem: As shown in the diagram, ∠EHC=35°, ∠GCH=28°, ∠HDF=25°, ∠HFG=51°, CE⊥HE, FG⊥HG, HC is perpendicular to AC. Find the measure of ∠GHF.
问题：如图所示，∠EHC=35°，∠GCH=28°，∠HDF=25°，∠HFG=51°，CE⊥HE，FG⊥HG，HC⊥AC。求∠GHF的大小。
Construction (CDL):
Shape(HD,DF,FH)
Shape(HF,FG,GH)
Shape(HG,GC,CH)
Shape(HC,CE,EH)
Shape(EC,CA,AE)
Collinear(DFGC)
Collinear(HEA)
Conditions (CDL):
Equal(MeasureOfAngle(EHC),35)
Equal(MeasureOfAngle(GCH),28)
Equal(MeasureOfAngle(HDF),25)
Equal(MeasureOfAngle(HFG),51)
PerpendicularBetweenLine(CE,HE)
PerpendicularBetweenLine(FG,HG)
PerpendicularBetweenLine(HC,AC)
Goal: Value(MeasureOfAngle(GHF))
Answer: 39
Theorem sequence:
triangle_property_angle_sum(1,HFG)
Problem: As shown in the diagram, ∠ADE=43°, AD is parallel to BC, DC∥AB. Find the measure of ∠ABC.
问题：如图所示，∠ADE=43°，AD平行于BC，DC∥AB。求∠ABC的大小。
Construction (CDL):
Shape(AB,BC,CD,DA)
Shape(AD,DE)
Collinear(EDC)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),43)
ParallelBetweenLine(AD,BC)
ParallelBetweenLine(DC,AB)
Goal: Value(MeasureOfAngle(ABC))
Answer: 137
Theorem sequence:
adjacent_complementary_angle(1,CDA,ADE)
parallelogram_judgment_parallel_and_parallel(1,ABCD)
parallelogram_property_opposite_angle_equal(1,BCDA)
Problem: As shown in the diagram, ∠BDE=35°, ∠CAE=28°, ∠EBD=75°. Find the measure of ∠BEA.
问题：如图所示，∠BDE=35°，∠CAE=28°，∠EBD=75°。求∠BEA的大小。
Construction (CDL):
Shape(BD,DE,EB)
Shape(EC,CA,AE)
Shape(BE,EA)
Shape(CE,ED)
Collinear(DEA)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(BDE),35)
Equal(MeasureOfAngle(CAE),28)
Equal(MeasureOfAngle(EBD),75)
Goal: Value(MeasureOfAngle(BEA))
Answer: 110
Theorem sequence:
angle_addition(1,DEB,BEA)
triangle_property_angle_sum(1,BDE)
flat_angle(1,DEA)
Problem: As shown in the diagram, ∠ABE=32°. Find the measure of ∠EDC.
问题：如图所示，∠ABE=32°。求∠EDC的大小。
Construction (CDL):
Shape(HCA,AE,EC)
Shape(HAB,BA)
Shape(HBD,DE,EB)
Shape(HDC,CD)
Shape(EA,AB,BE)
Shape(ED,DC,CE)
Collinear(AED)
Collinear(CEB)
Cocircular(H,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),32)
Goal: Value(MeasureOfAngle(EDC))
Answer: 32
Theorem sequence:
arc_property_circumference_angle_external(1,HCA,D)
arc_property_circumference_angle_external(1,HCA,B)
Problem: As shown in the diagram, AB=3, AC=5, AE=y, BE=x, CD=7/2, DE=3, EB∥DC. Find the length of line BE.
问题：如图所示，AB=3，AC=5，AE=y，BE=x，CD=7/2，DE=3，EB平行于DC。求直线BE的长度。
Construction (CDL):
Shape(BA,AE,EB)
Shape(BE,ED,DC,CB)
Collinear(ABC)
Collinear(AED)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AC),5)
Equal(LengthOfLine(AE),y)
Equal(LengthOfLine(BE),x)
Equal(LengthOfLine(CD),7/2)
Equal(LengthOfLine(DE),3)
ParallelBetweenLine(EB,DC)
Goal: Value(LengthOfLine(BE))
Answer: 21/10
Theorem sequence:
parallel_property_corresponding_angle(1,EB,DC,A)
similar_triangle_judgment_aa(1,BAE,CAD)
similar_triangle_property_line_ratio(1,EBA,DCA)
similar_triangle_property_line_ratio(1,AEB,ADC)
Problem: As shown in the diagram, CE=ED, EC=6, EF=5, FE=EA, quadrilateral CFDA is a rhombus. Find the area of CFDA.
问题：如图所示，CE=ED，EC=6，EF=5，FE=EA，CFDA是菱形。求四边形CFDA的面积。
Construction (CDL):
Shape(CF,FE,EC)
Shape(EF,FD,DE)
Shape(ED,DA,AE)
Shape(CE,EA,AC)
Collinear(FEA)
Collinear(CED)
Conditions (CDL):
Equal(LengthOfLine(CE),LengthOfLine(ED))
Equal(LengthOfLine(EC),6)
Equal(LengthOfLine(EF),5)
Equal(LengthOfLine(FE),LengthOfLine(EA))
Rhombus(CFDA)
Goal: Value(AreaOfQuadrilateral(CFDA))
Answer: 60
Theorem sequence:
line_addition(1,FE,EA)
line_addition(1,CE,ED)
kite_area_formula_diagonal(1,CFDA)
Problem: As shown in the diagram, AC=16, CD=23, ∠BDE=60°, quadrilateral BDCA is a parallelogram, DE is perpendicular to BE. Find the area of quadrilateral BDCA.
问题：如图所示，AC=16，CD=23，∠BDE=60°，BA和DC是平行四边形BDCA的一组对边，DE⊥BE。求BDCA的面积。
Construction (CDL):
Shape(BD,DE,EB)
Shape(BE,EC,CA,AB)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(AC),16)
Equal(LengthOfLine(CD),23)
Equal(MeasureOfAngle(BDE),60)
Parallelogram(BDCA)
PerpendicularBetweenLine(DE,BE)
Goal: Value(AreaOfQuadrilateral(BDCA))
Answer: 184*sqrt(3)
Theorem sequence:
parallelogram_property_opposite_line_equal(1,BDCA)
parallelogram_area_formula_sine(1,BDCA)