id int32 1 6.98k | annotation stringclasses 132
values | source stringlengths 7 17 | problem_level int32 0 28 | problem_text_cn stringlengths 20 201 | problem_text_en stringlengths 58 424 | problem_img stringlengths 5 8 | construction_cdl listlengths 1 28 | text_cdl listlengths 0 16 | image_cdl listlengths 0 16 | goal_cdl stringlengths 8 131 | problem_answer stringclasses 906
values | theorem_seqs listlengths 0 28 | theorem_seqs_dag_json stringlengths 13 3.3k | image imagewidth (px) 48 1.6k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
101 | XiaokaiZhang_2023-04-02 | Geometry3k-102 | 3 | 如图所示，KL=6，∠JKL=148°，⊙K的圆心为K。求扇形KLJ的面积。 | As shown in the diagram, KL=6, ∠JKL=148°, the center of ⊙K is K. Find the area of the sector KLJ. | 101.png | [
"Shape(KL,KLJ,JK)",
"Shape(KJ,KJL,LK)",
"Cocircular(K,JL)"
] | [
"Equal(LengthOfLine(KL),6)",
"Equal(MeasureOfAngle(JKL),148)",
"IsCentreOfCircle(K,K)"
] | [
"Equal(LengthOfLine(KL),6)",
"Equal(MeasureOfAngle(JKL),148)",
"IsCentreOfCircle(K,K)"
] | Value(AreaOfSector(KLJ)) | 74*pi/5 | [
"radius_of_circle_property_length_equal(1,KL,K)",
"arc_property_center_angle(1,KLJ,K)",
"sector_area_formula(1,KLJ)"
] | {"START": ["radius_of_circle_property_length_equal(1,KL,K)", "arc_property_center_angle(1,KLJ,K)", "sector_area_formula(1,KLJ)"]} | |
102 | XiaokaiZhang_2023-03-12 | Geometry3k-103 | 2 | 如图所示，∠KJL=2*x+27°，∠KLA=100°，∠LKJ=2*x-11°。求∠LKJ的大小。 | As shown in the diagram, ∠KJL=2*x+27°, ∠KLA=100°, ∠LKJ=2*x-11°. Find the measure of ∠LKJ. | 102.png | [
"Shape(KJ,JL,LK)",
"Shape(KL,LA)",
"Collinear(JLA)"
] | [
"Equal(MeasureOfAngle(KJL),2*x+27)",
"Equal(MeasureOfAngle(KLA),100)",
"Equal(MeasureOfAngle(LKJ),2*x-11)"
] | [
"Equal(MeasureOfAngle(KJL),2*x+27)",
"Equal(MeasureOfAngle(KLA),100)",
"Equal(MeasureOfAngle(LKJ),2*x-11)"
] | Value(MeasureOfAngle(LKJ)) | 31 | [
"adjacent_complementary_angle(1,JKL,KLA)",
"triangle_property_angle_sum(1,KJL)"
] | {"START": ["adjacent_complementary_angle(1,JKL,KLA)", "triangle_property_angle_sum(1,KJL)"]} | |
103 | XiaokaiZhang_2023-03-12 | Geometry3k-104 | 1 | 如图所示，AB=c，CA=b，CB=a，∠ABC=60°，∠CAB=30°，b=18，BC垂直于AC。求c的值。 | As shown in the diagram, AB=c, CA=b, CB=a, ∠ABC=60°, ∠CAB=30°, b=18, BC is perpendicular to AC. Find the value of c. | 103.png | [
"Shape(BC,CA,AB)"
] | [
"Equal(LengthOfLine(AB),c)",
"Equal(LengthOfLine(CA),b)",
"Equal(LengthOfLine(CB),a)",
"Equal(MeasureOfAngle(ABC),60)",
"Equal(MeasureOfAngle(CAB),30)",
"Equal(b,18)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),c)",
"Equal(LengthOfLine(CA),b)",
"Equal(LengthOfLine(CB),a)",
"Equal(MeasureOfAngle(ABC),60)",
"Equal(MeasureOfAngle(CAB),30)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(c) | 12*sqrt(3) | [
"sine_theorem(1,ABC)"
] | {"START": ["sine_theorem(1,ABC)"]} | |
104 | XiaokaiZhang_2023-03-12 | Geometry3k-105 | 1 | 如图所示，AB=x，AC=y，BC=6，∠BAC=30°，AC⊥BC。求x的值。 | As shown in the diagram, AB=x, AC=y, BC=6, ∠BAC=30°, AC is perpendicular to BC. Find the value of x. | 104.png | [
"Shape(CB,BA,AC)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),y)",
"Equal(LengthOfLine(BC),6)",
"Equal(MeasureOfAngle(BAC),30)",
"PerpendicularBetweenLine(AC,BC)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),y)",
"Equal(LengthOfLine(BC),6)",
"Equal(MeasureOfAngle(BAC),30)",
"PerpendicularBetweenLine(AC,BC)"
] | Value(x) | 12 | [
"sine_theorem(1,BAC)"
] | {"START": ["sine_theorem(1,BAC)"]} | |
105 | XiaokaiZhang_2023-03-12 | Geometry3k-106 | 1 | 如图所示，AD=15，PF=6，三角形ACE的重心为O。求直线PC的长度。 | As shown in the diagram, AD=15, PF=6, the centroid of triangle ACE is P. Find the length of line PC. | 105.png | [
"Shape(AB,BP,PA)",
"Shape(PB,BC,CP)",
"Shape(PC,CD,DP)",
"Shape(AP,PF,FA)",
"Shape(PD,DE,EP)",
"Shape(FP,PE,EF)",
"Collinear(ABC)",
"Collinear(CDE)",
"Collinear(EFA)",
"Collinear(BPE)",
"Collinear(CPF)",
"Collinear(APD)"
] | [
"Equal(LengthOfLine(AD),15)",
"Equal(LengthOfLine(PF),6)",
"IsCentroidOfTriangle(P,ACE)"
] | [] | Value(LengthOfLine(PC)) | 12 | [
"centroid_of_triangle_property_line_ratio(1,P,CEA,F)"
] | {"START": ["centroid_of_triangle_property_line_ratio(1,P,CEA,F)"]} | |
106 | XiaokaiZhang_2023-04-02 | Geometry3k-108 | 1 | 如图所示，ABCD的面积为846，EFGH的面积为376，CB=x，FG=24，ABCD相似于EFGH。求ABCD和EFGH的相似比。 | As shown in the diagram, the area of ABCD is 846, the area of quadrilateral EFGH is 376, CB=x, FG=24, ABCD is similar to EFGH. Find The ratio of similar quadrilaterals ABCD and EFGH. | 106.png | [
"Shape(DA,AB,BC,CD)",
"Shape(HE,EF,FG,GH)"
] | [
"Equal(AreaOfQuadrilateral(ABCD),846)",
"Equal(AreaOfQuadrilateral(EFGH),376)",
"Equal(LengthOfLine(CB),x)",
"Equal(LengthOfLine(FG),24)",
"SimilarBetweenQuadrilateral(ABCD,EFGH)"
] | [
"Equal(LengthOfLine(CB),x)",
"Equal(LengthOfLine(FG),24)"
] | Value(RatioOfSimilarQuadrilateral(ABCD,EFGH)) | 3/2 | [
"similar_quadrilateral_property_area_square_ratio(1,ABCD,EFGH)"
] | {"START": ["similar_quadrilateral_property_area_square_ratio(1,ABCD,EFGH)"]} | |
107 | XiaokaiZhang_2023-03-12 | Geometry3k-109 | 1 | 如图所示，AB=15*x+9，AC=7*x，BC=11*x+5，三角形CBA的周长为320。求直线CB的长度。 | As shown in the diagram, AB=15*x+9, AC=7*x, BC=11*x+5, the perimeter of triangle CBA is 320. Find the length of line CB. | 107.png | [
"Shape(CB,BA,AC)"
] | [
"Equal(LengthOfLine(AB),15*x+9)",
"Equal(LengthOfLine(AC),7*x)",
"Equal(LengthOfLine(BC),11*x+5)",
"Equal(PerimeterOfTriangle(CBA),320)"
] | [
"Equal(LengthOfLine(AB),15*x+9)",
"Equal(LengthOfLine(AC),7*x)",
"Equal(LengthOfLine(BC),11*x+5)",
"Equal(PerimeterOfTriangle(CBA),320)"
] | Value(LengthOfLine(CB)) | 107 | [
"triangle_perimeter_formula(1,CBA)"
] | {"START": ["triangle_perimeter_formula(1,CBA)"]} | |
108 | XiaokaiZhang_2023-04-02 | Geometry3k-110 | 2 | 如图所示，∠PRQ=115°，⊙R的圆心为R，NR垂直于PR。求⌒RMQ的角度。 | As shown in the diagram, ∠PRQ=115°, the center of circle R is R, NR is perpendicular to PR. Find the measure of ⌒RMQ. | 108.png | [
"Shape(RPN,NR,RP)",
"Shape(RNM,MR,RN)",
"Shape(RMQ,QR,RM)",
"Shape(RQP,PR,RQ)",
"Collinear(PRM)",
"Cocircular(R,PNMQ)"
] | [
"Equal(MeasureOfAngle(PRQ),115)",
"IsCentreOfCircle(R,R)",
"PerpendicularBetweenLine(NR,PR)"
] | [
"Equal(MeasureOfAngle(PRQ),115)",
"IsCentreOfCircle(R,R)",
"PerpendicularBetweenLine(NR,PR)"
] | Value(MeasureOfArc(RMQ)) | 65 | [
"adjacent_complementary_angle(1,PRQ,QRM)",
"arc_property_center_angle(1,RMQ,R)"
] | {"START": ["adjacent_complementary_angle(1,PRQ,QRM)", "arc_property_center_angle(1,RMQ,R)"]} | |
109 | XiaokaiZhang_2023-03-12 | Geometry3k-111 | 1 | 如图所示，AB=12，AC=y，BC=x，∠ACB=60°，∠BAC=30°，CB垂直于AB。求y的值。 | As shown in the diagram, AB=12, AC=y, BC=x, ∠ACB=60°, ∠BAC=30°, CB is perpendicular to AB. Find the value of y. | 109.png | [
"Shape(AC,CB,BA)"
] | [
"Equal(LengthOfLine(AB),12)",
"Equal(LengthOfLine(AC),y)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(ACB),60)",
"Equal(MeasureOfAngle(BAC),30)",
"PerpendicularBetweenLine(CB,AB)"
] | [
"Equal(LengthOfLine(AB),12)",
"Equal(LengthOfLine(AC),y)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(ACB),60)",
"Equal(MeasureOfAngle(BAC),30)",
"PerpendicularBetweenLine(CB,AB)"
] | Value(y) | 8*sqrt(3) | [
"sine_theorem(1,ACB)"
] | {"START": ["sine_theorem(1,ACB)"]} | |
110 | XiaokaiZhang_2023-04-02 | Geometry3k-112 | 1 | 如图所示，BO=15，BO=CA，CE=7，CO=13，CO=BA，OE垂直于AE。求四边形OCAB的周长。 | As shown in the diagram, BO=15, BO=CA, CE=7, CO=13, CO=BA, OE is perpendicular to AE. Find the perimeter of OCAB. | 110.png | [
"Shape(OC,CE,EO)",
"Shape(OE,EA,AB,BO)",
"Collinear(CEA)"
] | [
"Equal(LengthOfLine(BO),15)",
"Equal(LengthOfLine(BO),LengthOfLine(CA))",
"Equal(LengthOfLine(CE),7)",
"Equal(LengthOfLine(CO),13)",
"Equal(LengthOfLine(CO),LengthOfLine(BA))",
"PerpendicularBetweenLine(OE,AE)"
] | [
"Equal(LengthOfLine(BO),15)",
"Equal(LengthOfLine(BO),LengthOfLine(CA))",
"Equal(LengthOfLine(CE),7)",
"Equal(LengthOfLine(CO),13)",
"Equal(LengthOfLine(CO),LengthOfLine(BA))",
"PerpendicularBetweenLine(OE,AE)"
] | Value(PerimeterOfQuadrilateral(OCAB)) | 56 | [
"quadrilateral_perimeter_formula(1,OCAB)"
] | {"START": ["quadrilateral_perimeter_formula(1,OCAB)"]} | |
111 | XiaokaiZhang_2023-03-12 | Geometry3k-113 | 2 | 如图所示，∠ACD=50°，∠CDE=78°，∠FGA=120°，∠GFB=56°。求∠EAG的大小。 | As shown in the diagram, ∠ACD=50°, ∠CDE=78°, ∠FGA=120°, ∠GFB=56°. Find the measure of ∠EAG. | 111.png | [
"Shape(CD,DA,AC)",
"Shape(EA,AG,GE)",
"Shape(GF,FB,BG)",
"Shape(CA,AE)",
"Shape(EG,GB)",
"Shape(GA,AD)",
"Shape(FG,GA)",
"Collinear(CAGB)",
"Collinear(EAD)",
"Collinear(EGF)"
] | [
"Equal(MeasureOfAngle(ACD),50)",
"Equal(MeasureOfAngle(CDE),78)",
"Equal(MeasureOfAngle(FGA),120)",
"Equal(MeasureOfAngle(GFB),56)"
] | [
"Equal(MeasureOfAngle(ACD),50)",
"Equal(MeasureOfAngle(CDE),78)",
"Equal(MeasureOfAngle(FGA),120)",
"Equal(MeasureOfAngle(GFB),56)"
] | Value(MeasureOfAngle(EAG)) | 52 | [
"vertical_angle(1,EAG,DAC)",
"triangle_property_angle_sum(1,CDA)"
] | {"START": ["vertical_angle(1,EAG,DAC)", "triangle_property_angle_sum(1,CDA)"]} | |
112 | XiaokaiZhang_2023-04-02 | Geometry3k-114 | 2 | 如图所示，RQ=5，VS=11，S平分线段RT，V平分线段QU，QU和TR是梯形QUTR的腰。求直线UT的长度。 | As shown in the diagram, RQ=5, VS=11, S is the midpoint of segment RT, V bisects segment QU, QR and UT are the parallel sides of trapezoid QUTR. Find the length of line UT. | 112.png | [
"Shape(QV,VS,SR,RQ)",
"Shape(VU,UT,TS,SV)",
"Collinear(QVU)",
"Collinear(RST)"
] | [
"Equal(LengthOfLine(RQ),5)",
"Equal(LengthOfLine(VS),11)",
"IsMidpointOfLine(S,RT)",
"IsMidpointOfLine(V,QU)",
"Trapezoid(QUTR)"
] | [] | Value(LengthOfLine(UT)) | 17 | [
"midsegment_of_quadrilateral_judgment_midpoint(1,VS,QUTR)",
"midsegment_of_quadrilateral_property_length(1,VS,QUTR)"
] | {"START": ["midsegment_of_quadrilateral_judgment_midpoint(1,VS,QUTR)"], "midsegment_of_quadrilateral_judgment_midpoint(1,VS,QUTR)": ["midsegment_of_quadrilateral_property_length(1,VS,QUTR)"]} | |
113 | XiaokaiZhang_2023-04-02 | Geometry3k-115 | 1 | 如图所示，FK=3*x-1，JK=6*y-2，KG=4*y+3，KH=2*x+3，四边形FJHG是平行四边形。求x的值。 | As shown in the diagram, FK=3*x-1, JK=6*y-2, KG=4*y+3, KH=2*x+3, FG and JH are opposite sides of the ▱ FJHG. Find the value of x. | 113.png | [
"Shape(FJ,JK,KF)",
"Shape(KJ,JH,HK)",
"Shape(KH,HG,GK)",
"Shape(FK,KG,GF)",
"Collinear(FKH)",
"Collinear(JKG)"
] | [
"Equal(LengthOfLine(FK),3*x-1)",
"Equal(LengthOfLine(JK),6*y-2)",
"Equal(LengthOfLine(KG),4*y+3)",
"Equal(LengthOfLine(KH),2*x+3)",
"Parallelogram(FJHG)"
] | [] | Value(x) | 4 | [
"parallelogram_property_diagonal_bisection(1,FJHG,K)"
] | {"START": ["parallelogram_property_diagonal_bisection(1,FJHG,K)"]} | |
114 | XiaokaiZhang_2023-03-12 | Geometry3k-116 | 1 | 如图所示，AC=5，BC=x，∠ABC=60°，∠CAB=30°。求x的值。 | As shown in the diagram, AC=5, BC=x, ∠ABC=60°, ∠CAB=30°. Find the value of x. | 114.png | [
"Shape(BC,CA,AB)"
] | [
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(ABC),60)",
"Equal(MeasureOfAngle(CAB),30)"
] | [
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(ABC),60)",
"Equal(MeasureOfAngle(CAB),30)"
] | Value(x) | 5*sqrt(3)/3 | [
"sine_theorem(1,CAB)"
] | {"START": ["sine_theorem(1,CAB)"]} | |
115 | XiaokaiZhang_2023-04-02 | Geometry3k-117 | 2 | 如图所示，∠FOE=118°，∠LDA=104°，OD∥FI。求∠OFI的大小。 | As shown in the diagram, ∠FOE=118°, ∠LDA=104°, OD is parallel to FI. Find the measure of ∠OFI. | 115.png | [
"Shape(LD,DA)",
"Shape(AD,DI)",
"Shape(DI,IB)",
"Shape(BI,IC)",
"Shape(CI,IF)",
"Shape(IF,FJ)",
"Shape(JF,FK)",
"Shape(KF,FO)",
"Shape(FO,OE)",
"Shape(EO,OH)",
"Shape(HO,OD)",
"Shape(ID,DL)",
"Shape(DO,OF,FI,ID)",
"Collinear(LDIC)",
"Collinear(HOFJ)",
"Collinear(ADOE)",
"Collinear(BI... | [
"Equal(MeasureOfAngle(FOE),118)",
"Equal(MeasureOfAngle(LDA),104)",
"ParallelBetweenLine(OD,FI)"
] | [
"ParallelBetweenLine(OD,FI)"
] | Value(MeasureOfAngle(OFI)) | 118 | [
"adjacent_complementary_angle(1,DOF,FOE)",
"parallel_property_ipsilateral_internal_angle(1,OD,FI)"
] | {"START": ["adjacent_complementary_angle(1,DOF,FOE)", "parallel_property_ipsilateral_internal_angle(1,OD,FI)"]} | |
116 | XiaokaiZhang_2023-04-02 | Geometry3k-118 | 1 | 如图所示，∠GOI=3*y+1°，∠HBI=3*x+11°，∠OIE=4*x-5°，GE平行于OI，IB∥OH，OI∥HB。求x的值。 | As shown in the diagram, ∠GOI=3*y+1°, ∠HBI=3*x+11°, ∠OIE=4*x-5°, GE∥OI, IB is parallel to OH, OI∥HB. Find the value of x. | 116.png | [
"Shape(GO,OI,IE,EG)",
"Shape(OH,HB,BI,IO)",
"Collinear(GOH)",
"Collinear(BIE)"
] | [
"Equal(MeasureOfAngle(GOI),3*y+1)",
"Equal(MeasureOfAngle(HBI),3*x+11)",
"Equal(MeasureOfAngle(OIE),4*x-5)",
"ParallelBetweenLine(GE,OI)",
"ParallelBetweenLine(IB,OH)",
"ParallelBetweenLine(OI,HB)"
] | [
"Equal(MeasureOfAngle(GOI),3*y+1)",
"Equal(MeasureOfAngle(HBI),3*x+11)",
"Equal(MeasureOfAngle(OIE),4*x-5)",
"ParallelBetweenLine(GE,OI)",
"ParallelBetweenLine(IB,OH)",
"ParallelBetweenLine(OI,HB)"
] | Value(x) | 16 | [
"parallel_property_corresponding_angle(2,BH,IO,E)"
] | {"START": ["parallel_property_corresponding_angle(2,BH,IO,E)"]} | |
117 | XiaokaiZhang_2023-03-12 | Geometry3k-119 | 1 | 如图所示，AC=5*sqrt(26)，AD=25，CB=sqrt(26)，CD=5，DB=1，∠BCD=y°，∠DCA=x°，AD⊥CD，BC垂直于AC。求cos(CA)的值。 | As shown in the diagram, AC=5*sqrt(26), AD=25, CB=sqrt(26), CD=5, DB=1, ∠BCD=y°, ∠DCA=x°, AD is perpendicular to CD, BC is perpendicular to AC. Find the value of cos(CA). | 117.png | [
"Shape(CA,AD,DC)",
"Shape(CD,DB,BC)",
"Collinear(ADB)"
] | [
"Equal(LengthOfLine(AC),5*sqrt(26))",
"Equal(LengthOfLine(AD),25)",
"Equal(LengthOfLine(CB),sqrt(26))",
"Equal(LengthOfLine(CD),5)",
"Equal(LengthOfLine(DB),1)",
"Equal(MeasureOfAngle(BCD),y)",
"Equal(MeasureOfAngle(DCA),x)",
"PerpendicularBetweenLine(AD,CD)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AC),5*sqrt(26))",
"Equal(LengthOfLine(AD),25)",
"Equal(LengthOfLine(CB),sqrt(26))",
"Equal(LengthOfLine(CD),5)",
"Equal(LengthOfLine(DB),1)",
"Equal(MeasureOfAngle(BCD),y)",
"Equal(MeasureOfAngle(DCA),x)",
"PerpendicularBetweenLine(AD,CD)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(Cos(MeasureOfAngle(CAD))) | 5*sqrt(26)/26 | [
"cosine_theorem(1,ADC)"
] | {"START": ["cosine_theorem(1,ADC)"]} | |
118 | XiaokaiZhang_2023-04-02 | Geometry3k-120 | 1 | 如图所示，A是圆A的圆心，圆O的切线为DF。求∠AFD的大小。 | As shown in the diagram, A is the center of ⊙A, DF is the tangent to circle A. Find the measure of ∠AFD. | 118.png | [
"Shape(AF,AFB,BA)",
"Shape(AB,ABF,FA)",
"Shape(AF,FD)",
"Collinear(FABE)",
"Cocircular(A,FB)"
] | [
"IsCentreOfCircle(A,A)",
"IsTangentOfCircle(DF,A)"
] | [
"IsCentreOfCircle(A,A)",
"IsTangentOfCircle(DF,A)"
] | Value(MeasureOfAngle(AFD)) | 90 | [
"tangent_of_circle_property_perpendicular(1,DF,A,A)"
] | {"START": ["tangent_of_circle_property_perpendicular(1,DF,A,A)"]} | |
119 | XiaokaiZhang_2023-03-12 | Geometry3k-121 | 0 | 如图所示，WX=9*x，WY=6*x+3，WY=YX，XW=WY，YX=4*x+5。求直线XY的长度。 | As shown in the diagram, WX=9*x, WY=6*x+3, WY=YX, XW=WY, YX=4*x+5. Find the length of line XY. | 119.png | [
"Shape(XW,WY,YX)"
] | [
"Equal(LengthOfLine(WX),9*x)",
"Equal(LengthOfLine(WY),6*x+3)",
"Equal(LengthOfLine(WY),LengthOfLine(YX))",
"Equal(LengthOfLine(XW),LengthOfLine(WY))",
"Equal(LengthOfLine(YX),4*x+5)"
] | [
"Equal(LengthOfLine(WX),9*x)",
"Equal(LengthOfLine(WY),6*x+3)",
"Equal(LengthOfLine(WY),LengthOfLine(YX))",
"Equal(LengthOfLine(XW),LengthOfLine(WY))",
"Equal(LengthOfLine(YX),4*x+5)"
] | Value(LengthOfLine(XY)) | 9 | [] | {"START": []} | |
120 | XiaokaiZhang_2023-03-12 | Geometry3k-122 | 1 | 如图所示，ST=11*x-2，TU=8*x+4，UV=15*x，△RST与△VTU是镜像全等三角形，RS⊥TS，TU垂直于VU。求x的值。 | As shown in the diagram, ST=11*x-2, TU=8*x+4, UV=15*x, triangle RST is mirror congruent to triangle VTU, RS⊥TS, TU is perpendicular to VU. Find the value of x. | 120.png | [
"Shape(RS,ST,TR)",
"Shape(TU,UV,VT)",
"Collinear(STU)"
] | [
"Equal(LengthOfLine(ST),11*x-2)",
"Equal(LengthOfLine(TU),8*x+4)",
"Equal(LengthOfLine(UV),15*x)",
"MirrorCongruentBetweenTriangle(RST,VTU)",
"PerpendicularBetweenLine(RS,TS)",
"PerpendicularBetweenLine(TU,VU)"
] | [
"Equal(LengthOfLine(ST),11*x-2)",
"Equal(LengthOfLine(TU),8*x+4)",
"Equal(LengthOfLine(UV),15*x)",
"MirrorCongruentBetweenTriangle(RST,VTU)",
"PerpendicularBetweenLine(RS,TS)",
"PerpendicularBetweenLine(TU,VU)"
] | Value(x) | 2 | [
"mirror_congruent_triangle_property_line_equal(1,RST,VTU)"
] | {"START": ["mirror_congruent_triangle_property_line_equal(1,RST,VTU)"]} | |
121 | XiaokaiZhang_2023-04-02 | Geometry3k-123 | 3 | 如图所示，CB=20，EC=24，∠CAE=37°，ACBD是▱，AE垂直于CE。求ACBD的面积。 | As shown in the diagram, CB=20, EC=24, ∠CAE=37°, CA and BD are opposite sides of the ▱ ACBD, AE⊥CE. Find the area of quadrilateral ACBD. | 121.png | [
"Shape(EC,CA,AE)",
"Shape(AC,CB,BD,DA)",
"Collinear(EAD)"
] | [
"Equal(LengthOfLine(CB),20)",
"Equal(LengthOfLine(EC),24)",
"Equal(MeasureOfAngle(CAE),37)",
"Parallelogram(ACBD)",
"PerpendicularBetweenLine(AE,CE)"
] | [
"Equal(LengthOfLine(CB),20)",
"Equal(LengthOfLine(EC),24)",
"Equal(MeasureOfAngle(CAE),37)",
"PerpendicularBetweenLine(AE,CE)"
] | Value(AreaOfQuadrilateral(ACBD)) | 480 | [
"altitude_of_quadrilateral_judgment_right_vertex(5,CE,BDAC)",
"parallelogram_property_opposite_line_equal(1,CBDA)",
"parallelogram_area_formula_common(1,BDAC)"
] | {"START": ["altitude_of_quadrilateral_judgment_right_vertex(5,CE,BDAC)", "parallelogram_property_opposite_line_equal(1,CBDA)", "parallelogram_area_formula_common(1,BDAC)"]} | |
122 | XiaokaiZhang_2023-04-02 | Geometry3k-124 | 1 | 如图所示，CB=4*w-7，CD=11，ED=3*z+10，RS=2*w+13，RU=12，UT=z+16，∠CBE=2*x+9°，∠EDC=2*y-31°，∠STU=y+11°，∠URS=49°，BEDC与RSTU镜像全等。求x的值。 | As shown in the diagram, CB=4*w-7, CD=11, ED=3*z+10, RS=2*w+13, RU=12, UT=z+16, ∠CBE=2*x+9°, ∠EDC=2*y-31°, ∠STU=y+11°, ∠URS=49°, quadrilateral BEDC is mirror congruent to quadrilateral RSTU. Find the value of x. | 122.png | [
"Shape(BE,ED,DC,CB)",
"Shape(UR,RS,ST,TU)"
] | [
"Equal(LengthOfLine(CB),4*w-7)",
"Equal(LengthOfLine(CD),11)",
"Equal(LengthOfLine(ED),3*z+10)",
"Equal(LengthOfLine(RS),2*w+13)",
"Equal(LengthOfLine(RU),12)",
"Equal(LengthOfLine(UT),z+16)",
"Equal(MeasureOfAngle(CBE),2*x+9)",
"Equal(MeasureOfAngle(EDC),2*y-31)",
"Equal(MeasureOfAngle(STU),y+11)",... | [
"Equal(LengthOfLine(CB),4*w-7)",
"Equal(LengthOfLine(CD),11)",
"Equal(LengthOfLine(ED),3*z+10)",
"Equal(LengthOfLine(RS),2*w+13)",
"Equal(LengthOfLine(RU),12)",
"Equal(LengthOfLine(UT),z+16)",
"Equal(MeasureOfAngle(CBE),2*x+9)",
"Equal(MeasureOfAngle(EDC),2*y-31)",
"Equal(MeasureOfAngle(STU),y+11)",... | Value(x) | 20 | [
"mirror_congruent_quadrilateral_property_angle_equal(1,BEDC,RSTU)"
] | {"START": ["mirror_congruent_quadrilateral_property_angle_equal(1,BEDC,RSTU)"]} | |
123 | XiaokaiZhang_2023-03-12 | Geometry3k-125 | 3 | 如图所示，CB=12，CB=AB，∠BAC=44°。求∠CBA的大小。 | As shown in the diagram, CB=12, CB=AB, ∠BAC=44°. Find the measure of ∠CBA. | 123.png | [
"Shape(BA,AC,CB)"
] | [
"Equal(LengthOfLine(CB),12)",
"Equal(LengthOfLine(CB),LengthOfLine(AB))",
"Equal(MeasureOfAngle(BAC),44)"
] | [
"Equal(LengthOfLine(CB),12)",
"Equal(LengthOfLine(CB),LengthOfLine(AB))",
"Equal(MeasureOfAngle(BAC),44)"
] | Value(MeasureOfAngle(CBA)) | 92 | [
"isosceles_triangle_judgment_line_equal(1,BAC)",
"isosceles_triangle_property_angle_equal(1,BAC)",
"triangle_property_angle_sum(1,BAC)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,BAC)", "triangle_property_angle_sum(1,BAC)"], "isosceles_triangle_judgment_line_equal(1,BAC)": ["isosceles_triangle_property_angle_equal(1,BAC)"]} | |
124 | XiaokaiZhang_2023-04-02 | Geometry3k-126 | 4 | 如图所示，WT=3，WX⊥YX，XY垂直于ZY，YZ⊥WZ，ZW垂直于XW，XYZW是正方形。求直线XY的长度。 | As shown in the diagram, WT=3, WX is perpendicular to YX, XY⊥ZY, YZ is perpendicular to WZ, ZW⊥XW, quadrilateral XYZW is a square. Find the length of line XY. | 124.png | [
"Shape(XY,YT,TX)",
"Shape(TY,YZ,ZT)",
"Shape(TZ,ZW,WT)",
"Shape(XT,TW,WX)",
"Collinear(XTZ)",
"Collinear(WTY)"
] | [
"Equal(LengthOfLine(WT),3)",
"PerpendicularBetweenLine(WX,YX)",
"PerpendicularBetweenLine(XY,ZY)",
"PerpendicularBetweenLine(YZ,WZ)",
"PerpendicularBetweenLine(ZW,XW)",
"Square(XYZW)"
] | [
"PerpendicularBetweenLine(WX,YX)",
"PerpendicularBetweenLine(XY,ZY)",
"PerpendicularBetweenLine(YZ,WZ)",
"PerpendicularBetweenLine(ZW,XW)"
] | Value(LengthOfLine(XY)) | 3*sqrt(2) | [
"parallelogram_property_diagonal_bisection(1,YZWX,T)",
"line_addition(1,YT,TW)",
"right_triangle_judgment_angle(1,WXY)",
"right_triangle_property_pythagorean(1,WXY)"
] | {"START": ["parallelogram_property_diagonal_bisection(1,YZWX,T)", "line_addition(1,YT,TW)", "right_triangle_judgment_angle(1,WXY)"], "right_triangle_judgment_angle(1,WXY)": ["right_triangle_property_pythagorean(1,WXY)"]} | |
125 | XiaokaiZhang_2023-04-02 | Geometry3k-127 | 5 | 如图所示，∠ABH=3*p-10°，∠CDB=6*r+5°，∠JAB=4*p+15°，AJ平行于BH，CI平行于AL。求r的值。 | As shown in the diagram, ∠ABH=3*p-10°, ∠CDB=6*r+5°, ∠JAB=4*p+15°, AJ is parallel to BH, CI is parallel to AL. Find the value of r. | 125.png | [
"Shape(LA,AJ)",
"Shape(JA,AB)",
"Shape(AB,BH)",
"Shape(HB,BE)",
"Shape(EB,BD)",
"Shape(BD,DK)",
"Shape(KD,DF)",
"Shape(FD,DC)",
"Shape(DC,CG)",
"Shape(GC,CI)",
"Shape(IC,CA)",
"Shape(CA,AL)",
"Shape(AC,CD,DB,BA)",
"Collinear(LABE)",
"Collinear(ICDK)",
"Collinear(JACG)",
"Collinear(HB... | [
"Equal(MeasureOfAngle(ABH),3*p-10)",
"Equal(MeasureOfAngle(CDB),6*r+5)",
"Equal(MeasureOfAngle(JAB),4*p+15)",
"ParallelBetweenLine(AJ,BH)",
"ParallelBetweenLine(CI,AL)"
] | [
"ParallelBetweenLine(AJ,BH)",
"ParallelBetweenLine(CI,AL)"
] | Value(r) | 10 | [
"parallel_property_ipsilateral_internal_angle(1,AJ,BH)",
"adjacent_complementary_angle(1,DBA,ABH)",
"parallel_property_collinear_extend(2,LA,IC,B)",
"parallel_property_collinear_extend(1,CI,BA,D)",
"parallel_property_ipsilateral_internal_angle(1,DI,BA)"
] | {"START": ["parallel_property_ipsilateral_internal_angle(1,AJ,BH)", "adjacent_complementary_angle(1,DBA,ABH)", "parallel_property_collinear_extend(2,LA,IC,B)"], "parallel_property_collinear_extend(1,CI,BA,D)": ["parallel_property_ipsilateral_internal_angle(1,DI,BA)"], "parallel_property_collinear_extend(2,LA,IC,B)": ["... | |
126 | XiaokaiZhang_2023-03-12 | Geometry3k-128 | 3 | 如图所示，DA=3，DC=CB，DC⊥AC。求直线AB的长度。 | As shown in the diagram, DA=3, DC=CB, DC⊥AC. Find the length of line AB. | 126.png | [
"Shape(AD,DC,CA)",
"Shape(AC,CB,BA)",
"Collinear(DCB)"
] | [
"Equal(LengthOfLine(DA),3)",
"Equal(LengthOfLine(DC),LengthOfLine(CB))",
"PerpendicularBetweenLine(DC,AC)"
] | [
"Equal(LengthOfLine(DA),3)",
"Equal(LengthOfLine(DC),LengthOfLine(CB))",
"PerpendicularBetweenLine(DC,AC)"
] | Value(LengthOfLine(AB)) | 3 | [
"adjacent_complementary_angle(1,DCA,ACB)",
"mirror_congruent_triangle_judgment_sas(1,CAD,CBA)",
"mirror_congruent_triangle_property_line_equal(1,CAD,CBA)"
] | {"START": ["adjacent_complementary_angle(1,DCA,ACB)"], "adjacent_complementary_angle(1,DCA,ACB)": ["mirror_congruent_triangle_judgment_sas(1,CAD,CBA)"], "mirror_congruent_triangle_judgment_sas(1,CAD,CBA)": ["mirror_congruent_triangle_property_line_equal(1,CAD,CBA)"]} | |
127 | XiaokaiZhang_2023-04-02 | Geometry3k-129 | 2 | 如图所示，∠ADK=96°，∠HGJ=42°，GA∥HD。求∠GHD的大小。 | As shown in the diagram, ∠ADK=96°, ∠HGJ=42°, GA∥HD. Find the measure of ∠GHD. | 127.png | [
"Shape(KD,DL)",
"Shape(LD,DH)",
"Shape(DH,HI)",
"Shape(IH,HC)",
"Shape(CH,HG)",
"Shape(HG,GJ)",
"Shape(JG,GF)",
"Shape(FG,GA)",
"Shape(GA,AM)",
"Shape(MA,AF)",
"Shape(FA,AD)",
"Shape(AD,DK)",
"Shape(DA,AG,GH,HD)",
"Collinear(KDHC)",
"Collinear(EAGJ)",
"Collinear(LDAM)",
"Collinear(IH... | [
"Equal(MeasureOfAngle(ADK),96)",
"Equal(MeasureOfAngle(HGJ),42)",
"ParallelBetweenLine(GA,HD)"
] | [
"ParallelBetweenLine(GA,HD)"
] | Value(MeasureOfAngle(GHD)) | 42 | [
"parallel_property_collinear_extend(1,GA,HD,J)",
"parallel_property_alternate_interior_angle(2,DH,GJ)"
] | {"START": ["parallel_property_collinear_extend(1,GA,HD,J)"], "parallel_property_collinear_extend(1,GA,HD,J)": ["parallel_property_alternate_interior_angle(2,DH,GJ)"]} | |
128 | XiaokaiZhang_2023-04-02 | Geometry3k-130 | 1 | 如图所示，∠GKM=62°。求∠BKG的大小。 | As shown in the diagram, ∠GKM=62°. Find the measure of ∠BKG. | 128.png | [
"Shape(GK,KM)",
"Shape(BK,KG)",
"Collinear(MKB)"
] | [
"Equal(MeasureOfAngle(GKM),62)"
] | [
"Equal(MeasureOfAngle(GKM),62)"
] | Value(MeasureOfAngle(BKG)) | 118 | [
"adjacent_complementary_angle(1,BKG,GKM)"
] | {"START": ["adjacent_complementary_angle(1,BKG,GKM)"]} | |
129 | XiaokaiZhang_2023-03-12 | Geometry3k-131 | 1 | 如图所示，SR=5，TR=3，TS=4，RT⊥ST。求cos(TS)的值。 | As shown in the diagram, SR=5, TR=3, TS=4, RT is perpendicular to ST. Find the value of cos(TS). | 129.png | [
"Shape(TS,SR,RT)"
] | [
"Equal(LengthOfLine(SR),5)",
"Equal(LengthOfLine(TR),3)",
"Equal(LengthOfLine(TS),4)",
"PerpendicularBetweenLine(RT,ST)"
] | [
"Equal(LengthOfLine(SR),5)",
"Equal(LengthOfLine(TR),3)",
"Equal(LengthOfLine(TS),4)",
"PerpendicularBetweenLine(RT,ST)"
] | Value(Cos(MeasureOfAngle(TSR))) | 4/5 | [
"cosine_theorem(1,SRT)"
] | {"START": ["cosine_theorem(1,SRT)"]} | |
130 | XiaokaiZhang_2023-04-02 | Geometry3k-132 | 7 | 如图所示，AB=5*x，CA=3*x+4，WX=22，XY=22，⊙A的圆心为A，AB垂直于XB，XC垂直于AC。求直线AB的长度。 | As shown in the diagram, AB=5*x, CA=3*x+4, WX=22, XY=22, the center of circle A is A, AB⊥XB, XC is perpendicular to AC. Find the length of line AB. | 130.png | [
"Shape(AYW,WB,BA,AC,CY)",
"Shape(BW,AWX,XB)",
"Shape(AB,BX,XA)",
"Shape(AX,XC,CA)",
"Shape(CX,AXY,YC)",
"Collinear(WBX)",
"Collinear(XCY)",
"Cocircular(A,WXY)"
] | [
"Equal(LengthOfLine(AB),5*x)",
"Equal(LengthOfLine(CA),3*x+4)",
"Equal(LengthOfLine(WX),22)",
"Equal(LengthOfLine(XY),22)",
"IsCentreOfCircle(A,A)",
"PerpendicularBetweenLine(AB,XB)",
"PerpendicularBetweenLine(XC,AC)"
] | [
"Equal(LengthOfLine(AB),5*x)",
"Equal(LengthOfLine(CA),3*x+4)",
"IsCentreOfCircle(A,A)",
"PerpendicularBetweenLine(AB,XB)",
"PerpendicularBetweenLine(XC,AC)"
] | Value(LengthOfLine(AB)) | 10 | [
"adjacent_complementary_angle(1,WBA,ABX)",
"circle_property_chord_perpendicular_bisect_chord(1,A,AB,WX)",
"circle_property_chord_perpendicular_bisect_chord(1,A,AC,XY)",
"line_addition(1,WB,BX)",
"line_addition(1,XC,CY)",
"mirror_congruent_triangle_judgment_hl(1,ABX,AXC)",
"mirror_congruent_triangle_prop... | {"START": ["adjacent_complementary_angle(1,WBA,ABX)", "circle_property_chord_perpendicular_bisect_chord(1,A,AC,XY)", "line_addition(1,WB,BX)", "line_addition(1,XC,CY)"], "adjacent_complementary_angle(1,WBA,ABX)": ["circle_property_chord_perpendicular_bisect_chord(1,A,AB,WX)"], "circle_property_chord_perpendicular_bisec... | |
131 | XiaokaiZhang_2023-04-02 | Geometry3k-133 | 0 | 如图所示，AG=1/5*x+3，CJ=2*y+1，CJ=JE，EG=4*x-35，JE=5*y-8，AC平行于GJ。求y的值。 | As shown in the diagram, AG=1/5*x+3, CJ=2*y+1, CJ=JE, EG=4*x-35, JE=5*y-8, AC∥GJ. Find the value of y. | 131.png | [
"Shape(AG,GJ,JC,CA)",
"Shape(GE,EJ,JG)",
"Collinear(AGE)",
"Collinear(CJE)"
] | [
"Equal(LengthOfLine(AG),1/5*x+3)",
"Equal(LengthOfLine(CJ),2*y+1)",
"Equal(LengthOfLine(CJ),LengthOfLine(JE))",
"Equal(LengthOfLine(EG),4*x-35)",
"Equal(LengthOfLine(JE),5*y-8)",
"ParallelBetweenLine(AC,GJ)"
] | [
"Equal(LengthOfLine(AG),1/5*x+3)",
"Equal(LengthOfLine(CJ),2*y+1)",
"Equal(LengthOfLine(CJ),LengthOfLine(JE))",
"Equal(LengthOfLine(EG),4*x-35)",
"Equal(LengthOfLine(JE),5*y-8)",
"ParallelBetweenLine(AC,GJ)"
] | Value(y) | 3 | [] | {"START": []} | |
132 | XiaokaiZhang_2023-04-02 | Geometry3k-134 | 1 | 如图所示，AJ=2*x+3，BJ=5*x，JC=8*y-36，JD=4*y，ACBD是▱。求x的值。 | As shown in the diagram, AJ=2*x+3, BJ=5*x, JC=8*y-36, JD=4*y, CA and BD are opposite sides of the parallelogram ACBD. Find the value of x. | 132.png | [
"Shape(AC,CJ,JA)",
"Shape(JC,CB,BJ)",
"Shape(JB,BD,DJ)",
"Shape(AJ,JD,DA)",
"Collinear(AJB)",
"Collinear(CJD)"
] | [
"Equal(LengthOfLine(AJ),2*x+3)",
"Equal(LengthOfLine(BJ),5*x)",
"Equal(LengthOfLine(JC),8*y-36)",
"Equal(LengthOfLine(JD),4*y)",
"Parallelogram(ACBD)"
] | [
"Equal(LengthOfLine(AJ),2*x+3)",
"Equal(LengthOfLine(BJ),5*x)",
"Equal(LengthOfLine(JC),8*y-36)",
"Equal(LengthOfLine(JD),4*y)"
] | Value(x) | 1 | [
"parallelogram_property_diagonal_bisection(1,ACBD,J)"
] | {"START": ["parallelogram_property_diagonal_bisection(1,ACBD,J)"]} | |
133 | XiaokaiZhang_2023-04-02 | Geometry3k-135 | 3 | 如图所示，AB=36，AD=22，∠BCE=30°，BCDA是平行四边形，CE⊥BE。求BCDA的周长。 | As shown in the diagram, AB=36, AD=22, ∠BCE=30°, quadrilateral BCDA is a parallelogram, CE is perpendicular to BE. Find the perimeter of BCDA. | 133.png | [
"Shape(BC,CE,EB)",
"Shape(BE,ED,DA,AB)",
"Collinear(CED)"
] | [
"Equal(LengthOfLine(AB),36)",
"Equal(LengthOfLine(AD),22)",
"Equal(MeasureOfAngle(BCE),30)",
"Parallelogram(BCDA)",
"PerpendicularBetweenLine(CE,BE)"
] | [
"Equal(LengthOfLine(AB),36)",
"Equal(LengthOfLine(AD),22)",
"Equal(MeasureOfAngle(BCE),30)",
"PerpendicularBetweenLine(CE,BE)"
] | Value(PerimeterOfQuadrilateral(BCDA)) | 116 | [
"parallelogram_property_opposite_line_equal(1,BCDA)",
"parallelogram_property_opposite_line_equal(1,CDAB)",
"quadrilateral_perimeter_formula(1,BCDA)"
] | {"START": ["parallelogram_property_opposite_line_equal(1,BCDA)", "parallelogram_property_opposite_line_equal(1,CDAB)", "quadrilateral_perimeter_formula(1,BCDA)"]} | |
134 | XiaokaiZhang_2023-04-02 | Geometry3k-136 | 5 | 如图所示，CB=44，EA=19，∠ACE=30°，BDAC是平行四边形，DE垂直于CE。求BDAC的面积。 | As shown in the diagram, CB=44, EA=19, ∠ACE=30°, BC and DA are opposite sides of the ▱ BDAC, DE⊥CE. Find the area of quadrilateral BDAC. | 134.png | [
"Shape(BD,DE,EC,CB)",
"Shape(CE,EA,AC)",
"Collinear(DEA)"
] | [
"Equal(LengthOfLine(CB),44)",
"Equal(LengthOfLine(EA),19)",
"Equal(MeasureOfAngle(ACE),30)",
"Parallelogram(BDAC)",
"PerpendicularBetweenLine(DE,CE)"
] | [
"Equal(LengthOfLine(CB),44)",
"Equal(LengthOfLine(EA),19)",
"Equal(MeasureOfAngle(ACE),30)",
"PerpendicularBetweenLine(DE,CE)"
] | Value(AreaOfQuadrilateral(BDAC)) | 836*sqrt(3) | [
"triangle_property_angle_sum(1,CEA)",
"sine_theorem(1,EAC)",
"altitude_of_quadrilateral_judgment_right_vertex(1,CE,BDAC)",
"parallelogram_property_opposite_line_equal(1,DACB)",
"parallelogram_area_formula_common(1,BDAC)"
] | {"START": ["triangle_property_angle_sum(1,CEA)", "sine_theorem(1,EAC)", "altitude_of_quadrilateral_judgment_right_vertex(1,CE,BDAC)", "parallelogram_property_opposite_line_equal(1,DACB)", "parallelogram_area_formula_common(1,BDAC)"]} | |
135 | XiaokaiZhang_2023-04-02 | Geometry3k-137 | 2 | 如图所示，AC=25，AD=21，EB=20，CB和AD是平行四边形CADB的一组对边，DE⊥BE。求CADB的面积。 | As shown in the diagram, AC=25, AD=21, EB=20, quadrilateral CADB is a parallelogram, DE is perpendicular to BE. Find the area of CADB. | 135.png | [
"Shape(CA,AD,DB,BC)",
"Shape(BD,DE,EB)",
"Collinear(ADE)"
] | [
"Equal(LengthOfLine(AC),25)",
"Equal(LengthOfLine(AD),21)",
"Equal(LengthOfLine(EB),20)",
"Parallelogram(CADB)",
"PerpendicularBetweenLine(DE,BE)"
] | [
"Equal(LengthOfLine(AC),25)",
"Equal(LengthOfLine(AD),21)",
"Equal(LengthOfLine(EB),20)",
"PerpendicularBetweenLine(DE,BE)"
] | Value(AreaOfQuadrilateral(CADB)) | 420 | [
"altitude_of_quadrilateral_judgment_right_vertex(5,BE,CADB)",
"parallelogram_area_formula_common(1,CADB)"
] | {"START": ["altitude_of_quadrilateral_judgment_right_vertex(5,BE,CADB)", "parallelogram_area_formula_common(1,CADB)"]} | |
136 | XiaokaiZhang_2023-03-12 | Geometry3k-138 | 1 | 如图所示，AC=b，BA=c，BC=a，∠ABC=60°，∠CAB=30°，b=3，BC垂直于AC。求c的值。 | As shown in the diagram, AC=b, BA=c, BC=a, ∠ABC=60°, ∠CAB=30°, b=3, BC is perpendicular to AC. Find the value of c. | 136.png | [
"Shape(BC,CA,AB)"
] | [
"Equal(LengthOfLine(AC),b)",
"Equal(LengthOfLine(BA),c)",
"Equal(LengthOfLine(BC),a)",
"Equal(MeasureOfAngle(ABC),60)",
"Equal(MeasureOfAngle(CAB),30)",
"Equal(b,3)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AC),b)",
"Equal(LengthOfLine(BA),c)",
"Equal(LengthOfLine(BC),a)",
"Equal(MeasureOfAngle(ABC),60)",
"Equal(MeasureOfAngle(CAB),30)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(c) | 2*sqrt(3) | [
"sine_theorem(1,ABC)"
] | {"START": ["sine_theorem(1,ABC)"]} | |
137 | XiaokaiZhang_2023-04-02 | Geometry3k-139 | 3 | 如图所示，BC=3，∠CBA=10°，B是⊙B的圆心。求扇形BAC的面积。 | As shown in the diagram, BC=3, ∠CBA=10°, B is the center of ⊙B. Find the area of the sector BAC. | 137.png | [
"Shape(BAC,CB,BA)",
"Shape(BCA,AB,BC)",
"Cocircular(B,AC)"
] | [
"Equal(LengthOfLine(BC),3)",
"Equal(MeasureOfAngle(CBA),10)",
"IsCentreOfCircle(B,B)"
] | [
"Equal(LengthOfLine(BC),3)",
"Equal(MeasureOfAngle(CBA),10)",
"IsCentreOfCircle(B,B)"
] | Value(AreaOfSector(BAC)) | pi/4 | [
"arc_property_center_angle(1,BAC,B)",
"radius_of_circle_property_length_equal(1,BC,B)",
"sector_area_formula(1,BAC)"
] | {"START": ["arc_property_center_angle(1,BAC,B)", "radius_of_circle_property_length_equal(1,BC,B)", "sector_area_formula(1,BAC)"]} | |
138 | XiaokaiZhang_2023-04-02 | Geometry3k-140 | 2 | 如图所示，∠BCH=23°，∠HBC=32°，∠HID=22°。求∠BHD的大小。 | As shown in the diagram, ∠BCH=23°, ∠HBC=32°, ∠HID=22°. Find the measure of ∠BHD. | 138.png | [
"Shape(BC,CH,HB)",
"Shape(HI,ID,DH)",
"Shape(BH,HD)",
"Shape(FD,DI)",
"Shape(IH,HC)",
"Collinear(CHDE)"
] | [
"Equal(MeasureOfAngle(BCH),23)",
"Equal(MeasureOfAngle(HBC),32)",
"Equal(MeasureOfAngle(HID),22)"
] | [
"Equal(MeasureOfAngle(BCH),23)",
"Equal(MeasureOfAngle(HBC),32)",
"Equal(MeasureOfAngle(HID),22)"
] | Value(MeasureOfAngle(BHD)) | 55 | [
"triangle_property_angle_sum(1,BCH)",
"adjacent_complementary_angle(1,CHB,BHD)"
] | {"START": ["triangle_property_angle_sum(1,BCH)", "adjacent_complementary_angle(1,CHB,BHD)"]} | |
139 | XiaokaiZhang_2023-03-12 | Geometry3k-141 | 2 | 如图所示，JK=33，LK=x，∠KLJ=45°，JK⊥LK。求x的值。 | As shown in the diagram, JK=33, LK=x, ∠KLJ=45°, JK is perpendicular to LK. Find the value of x. | 139.png | [
"Shape(JK,KL,LJ)"
] | [
"Equal(LengthOfLine(JK),33)",
"Equal(LengthOfLine(LK),x)",
"Equal(MeasureOfAngle(KLJ),45)",
"PerpendicularBetweenLine(JK,LK)"
] | [
"Equal(LengthOfLine(JK),33)",
"Equal(LengthOfLine(LK),x)",
"Equal(MeasureOfAngle(KLJ),45)",
"PerpendicularBetweenLine(JK,LK)"
] | Value(x) | 33 | [
"triangle_property_angle_sum(1,JKL)",
"sine_theorem(1,KLJ)"
] | {"START": ["triangle_property_angle_sum(1,JKL)", "sine_theorem(1,KLJ)"]} | |
140 | XiaokaiZhang_2023-03-12 | Geometry3k-142 | 0 | 如图所示，JK=JL，JK=KL，JK=x+7，JL=4*x-8。求直线JK的长度。 | As shown in the diagram, JK=JL, JK=KL, JK=x+7, JL=4*x-8. Find the length of line JK. | 140.png | [
"Shape(KJ,JL,LK)"
] | [
"Equal(LengthOfLine(JK),LengthOfLine(JL))",
"Equal(LengthOfLine(JK),LengthOfLine(KL))",
"Equal(LengthOfLine(JK),x+7)",
"Equal(LengthOfLine(JL),4*x-8)"
] | [
"Equal(LengthOfLine(JK),LengthOfLine(JL))",
"Equal(LengthOfLine(JK),LengthOfLine(KL))",
"Equal(LengthOfLine(JK),x+7)",
"Equal(LengthOfLine(JL),4*x-8)"
] | Value(LengthOfLine(JK)) | 12 | [] | {"START": []} | |
141 | XiaokaiZhang_2023-04-02 | Geometry3k-143 | 0 | 如图所示，AB=1/4*x+5，BC=1/2*x-7，ED=66-2/3*y，FE=1/3*y-6，FE=ED，AB⊥EB，BC垂直于DC。求y的值。 | As shown in the diagram, AB=1/4*x+5, BC=1/2*x-7, ED=66-2/3*y, FE=1/3*y-6, FE=ED, AB is perpendicular to EB, BC⊥DC. Find the value of y. | 141.png | [
"Shape(AB,BE,EF,FA)",
"Shape(BC,CD,DE,EB)",
"Collinear(ABC)",
"Collinear(FED)"
] | [
"Equal(LengthOfLine(AB),1/4*x+5)",
"Equal(LengthOfLine(BC),1/2*x-7)",
"Equal(LengthOfLine(ED),66-2/3*y)",
"Equal(LengthOfLine(FE),1/3*y-6)",
"Equal(LengthOfLine(FE),LengthOfLine(ED))",
"PerpendicularBetweenLine(AB,EB)",
"PerpendicularBetweenLine(BC,DC)"
] | [
"Equal(LengthOfLine(AB),1/4*x+5)",
"Equal(LengthOfLine(BC),1/2*x-7)",
"Equal(LengthOfLine(ED),66-2/3*y)",
"Equal(LengthOfLine(FE),1/3*y-6)",
"Equal(LengthOfLine(FE),LengthOfLine(ED))",
"PerpendicularBetweenLine(AB,EB)",
"PerpendicularBetweenLine(BC,DC)"
] | Value(y) | 72 | [] | {"START": []} | |
142 | XiaokaiZhang_2023-03-12 | Geometry3k-144 | 0 | 如图所示，RS=3*x-5，RT=2*x+7，RT=RS，ST=22。求x的值。 | As shown in the diagram, RS=3*x-5, RT=2*x+7, RT=RS, ST=22. Find the value of x. | 142.png | [
"Shape(RS,ST,TR)"
] | [
"Equal(LengthOfLine(RS),3*x-5)",
"Equal(LengthOfLine(RT),2*x+7)",
"Equal(LengthOfLine(RT),LengthOfLine(RS))",
"Equal(LengthOfLine(ST),22)"
] | [
"Equal(LengthOfLine(RS),3*x-5)",
"Equal(LengthOfLine(RT),2*x+7)",
"Equal(LengthOfLine(RT),LengthOfLine(RS))",
"Equal(LengthOfLine(ST),22)"
] | Value(x) | 12 | [] | {"START": []} | |
143 | XiaokaiZhang_2023-04-02 | Geometry3k-145 | 0 | 如图所示，CB=6，∠DAB=Mul(MeasureOfAngle(CDA),2)°，ABCD是菱形。求直线DA的长度。 | As shown in the diagram, CB=6, ∠DAB=Mul(MeasureOfAngle(CDA),2)°, ABCD is a rhombus. Find the length of line DA. | 143.png | [
"Shape(DA,AE,ED)",
"Shape(EA,AB,BE)",
"Shape(DE,EC,CD)",
"Shape(CE,EB,BC)",
"Collinear(DEB)",
"Collinear(CEA)"
] | [
"Equal(LengthOfLine(CB),6)",
"Equal(MeasureOfAngle(DAB),Mul(MeasureOfAngle(CDA),2))",
"Rhombus(ABCD)"
] | [
"Equal(LengthOfLine(CB),6)",
"Equal(MeasureOfAngle(DAB),Mul(MeasureOfAngle(CDA),2))"
] | Value(LengthOfLine(DA)) | 6 | [] | {"START": []} | |
144 | XiaokaiZhang_2023-03-12 | Geometry3k-146 | 2 | 如图所示，AB=x，AC=x，BC=5*sqrt(2)，CA⊥BA。求x的值。 | As shown in the diagram, AB=x, AC=x, BC=5*sqrt(2), CA is perpendicular to BA. Find the value of x. | 144.png | [
"Shape(AB,BC,CA)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(BC),5*sqrt(2))",
"PerpendicularBetweenLine(CA,BA)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(BC),5*sqrt(2))",
"PerpendicularBetweenLine(CA,BA)"
] | Value(x) | 5 | [
"right_triangle_judgment_angle(1,CAB)",
"right_triangle_property_pythagorean(1,CAB)"
] | {"START": ["right_triangle_judgment_angle(1,CAB)"], "right_triangle_judgment_angle(1,CAB)": ["right_triangle_property_pythagorean(1,CAB)"]} | |
145 | XiaokaiZhang_2023-03-12 | Geometry3k-147 | 1 | 如图所示，AC=b，BA=c，BC=a，a=14，b=48，c=50，BC垂直于AC。求tan(AB)的值。 | As shown in the diagram, AC=b, BA=c, BC=a, a=14, b=48, c=50, BC is perpendicular to AC. Find the value of tan(AB). | 145.png | [
"Shape(BC,CA,AB)"
] | [
"Equal(LengthOfLine(AC),b)",
"Equal(LengthOfLine(BA),c)",
"Equal(LengthOfLine(BC),a)",
"Equal(a,14)",
"Equal(b,48)",
"Equal(c,50)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AC),b)",
"Equal(LengthOfLine(BA),c)",
"Equal(LengthOfLine(BC),a)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(Tan(MeasureOfAngle(ABC))) | 24/7 | [
"cosine_theorem(1,BCA)"
] | {"START": ["cosine_theorem(1,BCA)"]} | |
146 | XiaokaiZhang_2023-04-02 | Geometry3k-148 | 3 | 如图所示，BA=5，∠ABC=46°，B是⊙B的圆心。求扇形BCA的面积。 | As shown in the diagram, BA=5, ∠ABC=46°, B is the center of circle B. Find the area of the sector BCA. | 146.png | [
"Shape(BCA,AB,BC)",
"Shape(BA,BAC,CB)",
"Cocircular(B,CA)"
] | [
"Equal(LengthOfLine(BA),5)",
"Equal(MeasureOfAngle(ABC),46)",
"IsCentreOfCircle(B,B)"
] | [
"Equal(LengthOfLine(BA),5)",
"Equal(MeasureOfAngle(ABC),46)",
"IsCentreOfCircle(B,B)"
] | Value(AreaOfSector(BCA)) | 115*pi/36 | [
"radius_of_circle_property_length_equal(1,BA,B)",
"arc_property_center_angle(1,BCA,B)",
"sector_area_formula(1,BCA)"
] | {"START": ["radius_of_circle_property_length_equal(1,BA,B)", "arc_property_center_angle(1,BCA,B)", "sector_area_formula(1,BCA)"]} | |
147 | XiaokaiZhang_2023-03-12 | Geometry3k-149 | 1 | 如图所示，∠MLN=31°，∠QPN=22°，NM垂直于LM。求∠LNM的大小。 | As shown in the diagram, ∠MLN=31°, ∠QPN=22°, NM⊥LM. Find the measure of ∠LNM. | 147.png | [
"Shape(LN,NM,ML)",
"Shape(NQ,QP,PN)",
"Collinear(MNQ)",
"Collinear(LNP)"
] | [
"Equal(MeasureOfAngle(MLN),31)",
"Equal(MeasureOfAngle(QPN),22)",
"PerpendicularBetweenLine(NM,LM)"
] | [
"Equal(MeasureOfAngle(MLN),31)",
"Equal(MeasureOfAngle(QPN),22)",
"PerpendicularBetweenLine(NM,LM)"
] | Value(MeasureOfAngle(LNM)) | 59 | [
"triangle_property_angle_sum(1,LNM)"
] | {"START": ["triangle_property_angle_sum(1,LNM)"]} | |
148 | XiaokaiZhang_2023-03-12 | Geometry3k-150 | 1 | 如图所示，SR=5，TR=3，TS=4，RT垂直于ST。求sin(SR)的值。 | As shown in the diagram, SR=5, TR=3, TS=4, RT⊥ST. Find the value of sin(SR). | 148.png | [
"Shape(TS,SR,RT)"
] | [
"Equal(LengthOfLine(SR),5)",
"Equal(LengthOfLine(TR),3)",
"Equal(LengthOfLine(TS),4)",
"PerpendicularBetweenLine(RT,ST)"
] | [
"Equal(LengthOfLine(SR),5)",
"Equal(LengthOfLine(TR),3)",
"Equal(LengthOfLine(TS),4)",
"PerpendicularBetweenLine(RT,ST)"
] | Value(Sin(MeasureOfAngle(SRT))) | 4/5 | [
"cosine_theorem(1,RTS)"
] | {"START": ["cosine_theorem(1,RTS)"]} | |
149 | XiaokaiZhang_2023-03-12 | Geometry3k-151 | 1 | 如图所示，BA=3，BA=AC，BC=4*y-5，∠BAC=2*x°，∠CBA=2*x°。求y的值。 | As shown in the diagram, BA=3, BA=AC, BC=4*y-5, ∠BAC=2*x°, ∠CBA=2*x°. Find the value of y. | 149.png | [
"Shape(BA,AC,CB)"
] | [
"Equal(LengthOfLine(BA),3)",
"Equal(LengthOfLine(BA),LengthOfLine(AC))",
"Equal(LengthOfLine(BC),4*y-5)",
"Equal(MeasureOfAngle(BAC),2*x)",
"Equal(MeasureOfAngle(CBA),2*x)"
] | [
"Equal(LengthOfLine(BA),3)",
"Equal(LengthOfLine(BA),LengthOfLine(AC))",
"Equal(LengthOfLine(BC),4*y-5)",
"Equal(MeasureOfAngle(BAC),2*x)",
"Equal(MeasureOfAngle(CBA),2*x)"
] | Value(y) | 2 | [
"isosceles_triangle_judgment_angle_equal(1,CBA)"
] | {"START": ["isosceles_triangle_judgment_angle_equal(1,CBA)"]} | |
150 | XiaokaiZhang_2023-04-02 | Geometry3k-152 | 0 | 如图所示，AF=1/2*y+20，CF=3*y，CG=20-3*x，FC=AF，HG=2*x-5，FG∥AH。求y的值。 | As shown in the diagram, AF=1/2*y+20, CF=3*y, CG=20-3*x, FC=AF, HG=2*x-5, FG∥AH. Find the value of y. | 150.png | [
"Shape(CF,FG,GC)",
"Shape(FA,AH,HG,GF)",
"Collinear(CFA)",
"Collinear(CGH)"
] | [
"Equal(LengthOfLine(AF),1/2*y+20)",
"Equal(LengthOfLine(CF),3*y)",
"Equal(LengthOfLine(CG),20-3*x)",
"Equal(LengthOfLine(FC),LengthOfLine(AF))",
"Equal(LengthOfLine(HG),2*x-5)",
"ParallelBetweenLine(FG,AH)"
] | [
"Equal(LengthOfLine(AF),1/2*y+20)",
"Equal(LengthOfLine(CF),3*y)",
"Equal(LengthOfLine(CG),20-3*x)",
"Equal(LengthOfLine(FC),LengthOfLine(AF))",
"Equal(LengthOfLine(HG),2*x-5)",
"ParallelBetweenLine(FG,AH)"
] | Value(y) | 8 | [] | {"START": []} | |
151 | XiaokaiZhang_2023-04-02 | Geometry3k-153 | 1 | 如图所示，∠BCA=5*x°，∠DCB=3*x°。求x的值。 | As shown in the diagram, ∠BCA=5*x°, ∠DCB=3*x°. Find the value of x. | 151.png | [
"Shape(DC,CB)",
"Shape(BC,CA)",
"Collinear(DCA)"
] | [
"Equal(MeasureOfAngle(BCA),5*x)",
"Equal(MeasureOfAngle(DCB),3*x)"
] | [
"Equal(MeasureOfAngle(BCA),5*x)",
"Equal(MeasureOfAngle(DCB),3*x)"
] | Value(x) | 45/2 | [
"adjacent_complementary_angle(1,DCB,BCA)"
] | {"START": ["adjacent_complementary_angle(1,DCB,BCA)"]} | |
152 | XiaokaiZhang_2023-04-02 | Geometry3k-154 | 2 | 如图所示，∠ACF=140°，∠IGE=9*x°，GD平行于CA。求x的值。 | As shown in the diagram, ∠ACF=140°, ∠IGE=9*x°, GD is parallel to CA. Find the value of x. | 152.png | [
"Shape(EG,GD)",
"Shape(DG,GC)",
"Shape(GC,CA)",
"Shape(AC,CF)",
"Shape(FC,CB)",
"Shape(BC,CG)",
"Shape(CG,GI)",
"Shape(IG,GE)",
"Collinear(IGD)",
"Collinear(BCA)",
"Collinear(EGCF)"
] | [
"Equal(MeasureOfAngle(ACF),140)",
"Equal(MeasureOfAngle(IGE),9*x)",
"ParallelBetweenLine(GD,CA)"
] | [
"Equal(MeasureOfAngle(ACF),140)",
"Equal(MeasureOfAngle(IGE),9*x)",
"ParallelBetweenLine(GD,CA)"
] | Value(x) | 140/9 | [
"parallel_property_corresponding_angle(2,GD,CA,F)",
"vertical_angle(1,IGE,DGC)"
] | {"START": ["parallel_property_corresponding_angle(2,GD,CA,F)", "vertical_angle(1,IGE,DGC)"]} | |
153 | XiaokaiZhang_2023-04-02 | Geometry3k-155 | 1 | 如图所示，∠DAF=4*x+5°，∠FAB=9*x+20°，∠FBC=4*y+4°，∠FDA=y**2-1°，ABCD是矩形。求y的值。 | As shown in the diagram, ∠DAF=4*x+5°, ∠FAB=9*x+20°, ∠FBC=4*y+4°, ∠FDA=y**2-1°, quadrilateral ABCD is a rectangle. Find the value of y. | 153.png | [
"Shape(AB,BF,FA)",
"Shape(FB,BC,CF)",
"Shape(FC,CD,DF)",
"Shape(AF,FD,DA)",
"Collinear(AFC)",
"Collinear(BFD)"
] | [
"Equal(MeasureOfAngle(DAF),4*x+5)",
"Equal(MeasureOfAngle(FAB),9*x+20)",
"Equal(MeasureOfAngle(FBC),4*y+4)",
"Equal(MeasureOfAngle(FDA),y**2-1)",
"Rectangle(ABCD)"
] | [
"Equal(MeasureOfAngle(DAF),4*x+5)",
"Equal(MeasureOfAngle(FAB),9*x+20)",
"Equal(MeasureOfAngle(FBC),4*y+4)",
"Equal(MeasureOfAngle(FDA),y**2-1)"
] | Value(y) | 5 | [
"parallel_property_alternate_interior_angle(2,AD,BC)"
] | {"START": ["parallel_property_alternate_interior_angle(2,AD,BC)"]} | |
154 | XiaokaiZhang_2023-04-02 | Geometry3k-156 | 3 | 如图所示，AH=RH，HA=6-x，IJ=4/3*y+1，JE=2*y，RH=2*x+3，HJ平行于AE，RI∥HJ。求y的值。 | As shown in the diagram, AH=RH, HA=6-x, IJ=4/3*y+1, JE=2*y, RH=2*x+3, HJ is parallel to AE, RI is parallel to HJ. Find the value of y. | 154.png | [
"Shape(RH,HJ,JI,IR)",
"Shape(HA,AE,EJ,JH)",
"Collinear(RHA)",
"Collinear(IJE)"
] | [
"Equal(LengthOfLine(AH),LengthOfLine(RH))",
"Equal(LengthOfLine(HA),6-x)",
"Equal(LengthOfLine(IJ),4/3*y+1)",
"Equal(LengthOfLine(JE),2*y)",
"Equal(LengthOfLine(RH),2*x+3)",
"ParallelBetweenLine(HJ,AE)",
"ParallelBetweenLine(RI,HJ)"
] | [
"Equal(LengthOfLine(AH),LengthOfLine(RH))",
"Equal(LengthOfLine(HA),6-x)",
"Equal(LengthOfLine(IJ),4/3*y+1)",
"Equal(LengthOfLine(JE),2*y)",
"Equal(LengthOfLine(RH),2*x+3)",
"ParallelBetweenLine(HJ,AE)",
"ParallelBetweenLine(RI,HJ)"
] | Value(y) | 3/2 | [
"parallel_judgment_par_par(1,RI,HJ,AE)",
"trapezoid_judgment_parallel(1,RAEI)",
"midsegment_of_quadrilateral_judgment_parallel(1,HJ,RAEI)"
] | {"START": ["parallel_judgment_par_par(1,RI,HJ,AE)"], "parallel_judgment_par_par(1,RI,HJ,AE)": ["trapezoid_judgment_parallel(1,RAEI)"], "trapezoid_judgment_parallel(1,RAEI)": ["midsegment_of_quadrilateral_judgment_parallel(1,HJ,RAEI)"]} | |
155 | XiaokaiZhang_2023-04-02 | Geometry3k-157 | 16 | 如图所示，AD=27，BA=CD，CH=7，∠ABC=135°，IH∥BC，CH⊥IH，HI垂直于BI。求四边形ABCD的周长。 | As shown in the diagram, AD=27, BA=CD, CH=7, ∠ABC=135°, IH is parallel to BC, CH is perpendicular to IH, HI⊥BI. Find the perimeter of ABCD. | 155.png | [
"Shape(AB,BI,IA)",
"Shape(IB,BC,CH,HI)",
"Shape(HC,CD,DH)",
"Collinear(AIHD)"
] | [
"Equal(LengthOfLine(AD),27)",
"Equal(LengthOfLine(BA),LengthOfLine(CD))",
"Equal(LengthOfLine(CH),7)",
"Equal(MeasureOfAngle(ABC),135)",
"ParallelBetweenLine(IH,BC)",
"PerpendicularBetweenLine(CH,IH)",
"PerpendicularBetweenLine(HI,BI)"
] | [
"Equal(LengthOfLine(AD),27)",
"Equal(LengthOfLine(BA),LengthOfLine(CD))",
"Equal(LengthOfLine(CH),7)",
"Equal(MeasureOfAngle(ABC),135)",
"ParallelBetweenLine(IH,BC)",
"PerpendicularBetweenLine(CH,IH)",
"PerpendicularBetweenLine(HI,BI)"
] | Value(PerimeterOfQuadrilateral(ABCD)) | 14*sqrt(2)+40 | [
"adjacent_complementary_angle(1,HIB,BIA)",
"adjacent_complementary_angle(1,DHC,CHA)",
"parallel_judgment_ipsilateral_internal_angle(1,HC,IB)",
"parallelogram_judgment_parallel_and_parallel(1,IBCH)",
"parallelogram_property_opposite_line_equal(1,IBCH)",
"parallel_property_ipsilateral_internal_angle(1,IH,BC... | {"START": ["adjacent_complementary_angle(1,HIB,BIA)", "adjacent_complementary_angle(1,DHC,CHA)", "parallel_judgment_ipsilateral_internal_angle(1,HC,IB)", "parallel_property_ipsilateral_internal_angle(1,IH,BC)", "angle_addition(1,ABI,IBC)", "triangle_property_angle_sum(1,ABI)", "sine_theorem(1,IAB)", "sine_theorem(1,BIA... | |
156 | XiaokaiZhang_2023-04-02 | Geometry3k-158 | 1 | 如图所示，∠WZY=4*x°，∠XWZ=3*x°，∠YXW=x°，∠ZYX=2*x°。求∠ZYX的大小。 | As shown in the diagram, ∠WZY=4*x°, ∠XWZ=3*x°, ∠YXW=x°, ∠ZYX=2*x°. Find the measure of ∠ZYX. | 156.png | [
"Shape(XW,WZ,ZY,YX)"
] | [
"Equal(MeasureOfAngle(WZY),4*x)",
"Equal(MeasureOfAngle(XWZ),3*x)",
"Equal(MeasureOfAngle(YXW),x)",
"Equal(MeasureOfAngle(ZYX),2*x)"
] | [
"Equal(MeasureOfAngle(WZY),4*x)",
"Equal(MeasureOfAngle(XWZ),3*x)",
"Equal(MeasureOfAngle(YXW),x)",
"Equal(MeasureOfAngle(ZYX),2*x)"
] | Value(MeasureOfAngle(ZYX)) | 72 | [
"quadrilateral_property_angle_sum(1,XWZY)"
] | {"START": ["quadrilateral_property_angle_sum(1,XWZY)"]} | |
157 | XiaokaiZhang_2023-04-02 | Geometry3k-159 | 5 | 如图所示，BC=CE，CE=4*sqrt(2)，⊙D的圆心为D，BE是圆D的直径，EC垂直于BC。求圆D的周长。 | As shown in the diagram, BC=CE, CE=4*sqrt(2), the center of ⊙D is D, BE is the diameter of circle D, EC is perpendicular to BC. Find the circumference of the ⊙D. | 157.png | [
"Shape(DCB,BC)",
"Shape(DEC,CE)",
"Shape(CB,BD,DE,EC)",
"Shape(DB,DBE,ED)",
"Collinear(BDE)",
"Cocircular(D,CBE)"
] | [
"Equal(LengthOfLine(BC),LengthOfLine(CE))",
"Equal(LengthOfLine(CE),4*sqrt(2))",
"IsCentreOfCircle(D,D)",
"IsDiameterOfCircle(BE,D)",
"PerpendicularBetweenLine(EC,BC)"
] | [
"Equal(LengthOfLine(BC),LengthOfLine(CE))",
"Equal(LengthOfLine(CE),4*sqrt(2))",
"IsCentreOfCircle(D,D)",
"IsDiameterOfCircle(BE,D)",
"PerpendicularBetweenLine(EC,BC)"
] | Value(PerimeterOfCircle(D)) | 8*pi | [
"right_triangle_judgment_angle(1,ECB)",
"right_triangle_property_pythagorean(1,ECB)",
"diameter_of_circle_property_length_equal(1,BE,D)",
"circle_property_length_of_radius_and_diameter(1,D)",
"circle_perimeter_formula(1,D)"
] | {"START": ["right_triangle_judgment_angle(1,ECB)", "diameter_of_circle_property_length_equal(1,BE,D)", "circle_property_length_of_radius_and_diameter(1,D)", "circle_perimeter_formula(1,D)"], "right_triangle_judgment_angle(1,ECB)": ["right_triangle_property_pythagorean(1,ECB)"]} | |
158 | XiaokaiZhang_2023-04-02 | Geometry3k-160 | 3 | 如图所示，∠HPM=4*y°，∠MPR=68°，∠PRC=x°，∠SCR=5*z+2°，MC平行于PR，PM平行于RC。求z的值。 | As shown in the diagram, ∠HPM=4*y°, ∠MPR=68°, ∠PRC=x°, ∠SCR=5*z+2°, MC is parallel to PR, PM is parallel to RC. Find the value of z. | 158.png | [
"Shape(NC,CS)",
"Shape(SC,CR)",
"Shape(CR,RG)",
"Shape(GR,RI)",
"Shape(IR,RP)",
"Shape(RP,PL)",
"Shape(LP,PH)",
"Shape(HP,PM)",
"Shape(PM,ME)",
"Shape(EM,MD)",
"Shape(DM,MC)",
"Shape(MC,CN)",
"Shape(CM,MP,PR,RC)",
"Collinear(NCRI)",
"Collinear(DMPL)",
"Collinear(SCME)",
"Collinear(GR... | [
"Equal(MeasureOfAngle(HPM),4*y)",
"Equal(MeasureOfAngle(MPR),68)",
"Equal(MeasureOfAngle(PRC),x)",
"Equal(MeasureOfAngle(SCR),5*z+2)",
"ParallelBetweenLine(MC,PR)",
"ParallelBetweenLine(PM,RC)"
] | [
"Equal(MeasureOfAngle(HPM),4*y)",
"Equal(MeasureOfAngle(MPR),68)",
"Equal(MeasureOfAngle(PRC),x)",
"Equal(MeasureOfAngle(SCR),5*z+2)",
"ParallelBetweenLine(MC,PR)",
"ParallelBetweenLine(PM,RC)"
] | Value(z) | 22 | [
"parallelogram_judgment_parallel_and_parallel(1,CMPR)",
"parallelogram_property_opposite_angle_equal(1,CMPR)",
"adjacent_complementary_angle(1,SCR,RCM)"
] | {"START": ["parallelogram_judgment_parallel_and_parallel(1,CMPR)", "adjacent_complementary_angle(1,SCR,RCM)"], "parallelogram_judgment_parallel_and_parallel(1,CMPR)": ["parallelogram_property_opposite_angle_equal(1,CMPR)"]} | |
159 | XiaokaiZhang_2023-04-02 | Geometry3k-161 | 2 | 如图所示，∠DCG=3*x°，∠GBA=x+24°。求∠GBA的大小。 | As shown in the diagram, ∠DCG=3*x°, ∠GBA=x+24°. Find the measure of ∠GBA. | 159.png | [
"Shape(OBA,AB)",
"Shape(GB,BA,AG)",
"Shape(OAD,DG,GA)",
"Shape(GD,DC,CG)",
"Shape(ODC,CD)",
"Shape(OCB.BG,GC)",
"Collinear(BGD)",
"Collinear(CGA)",
"Cocircular(O,BADC)"
] | [
"Equal(MeasureOfAngle(DCG),3*x)",
"Equal(MeasureOfAngle(GBA),x+24)"
] | [
"Equal(MeasureOfAngle(DCG),3*x)",
"Equal(MeasureOfAngle(GBA),x+24)"
] | Value(MeasureOfAngle(GBA)) | 36 | [
"arc_property_circumference_angle_external(1,OAD,B)",
"arc_property_circumference_angle_external(1,OAD,C)"
] | {"START": ["arc_property_circumference_angle_external(1,OAD,B)", "arc_property_circumference_angle_external(1,OAD,C)"]} | |
160 | XiaokaiZhang_2023-04-02 | Geometry3k-162 | 2 | 如图所示，∠CBD=55°，∠FBG=35°，⊙B的圆心为B。求⌒BFG的角度。 | As shown in the diagram, ∠CBD=55°, ∠FBG=35°, B is the center of ⊙B. Find the measure of ⌒BFG. | 160.png | [
"Shape(BC,BCA,AB)",
"Shape(BA,BAG,GB)",
"Shape(BG,BGF,FB)",
"Shape(BF,BFD,DB)",
"Shape(BD,BDC,DB)",
"Collinear(CBG)",
"Collinear(ABD)",
"Cocircular(B,AGFDC)"
] | [
"Equal(MeasureOfAngle(CBD),55)",
"Equal(MeasureOfAngle(FBG),35)",
"IsCentreOfCircle(B,B)"
] | [
"Equal(MeasureOfAngle(CBD),55)",
"Equal(MeasureOfAngle(FBG),35)",
"IsCentreOfCircle(B,B)"
] | Value(MeasureOfArc(BFG)) | 325 | [
"round_angle(1,FBG,GBF)",
"arc_property_center_angle(1,BFG,B)"
] | {"START": ["round_angle(1,FBG,GBF)", "arc_property_center_angle(1,BFG,B)"]} | |
161 | XiaokaiZhang_2023-04-02 | Geometry3k-163 | 8 | 如图所示，AE=6，DH=6，EF=6，AB⊥FB，DH⊥BH，EA垂直于BA，FE⊥AE。求三角形DFB的面积与FEAB的面积之和。 | As shown in the diagram, AE=6, DH=6, EF=6, AB is perpendicular to FB, DH is perpendicular to BH, EA is perpendicular to BA, FE is perpendicular to AE. Find the sum of the area of triangle DFB and the area of FEAB. | 161.png | [
"Shape(DF,FH,HD)",
"Shape(DH,HB,BD)",
"Shape(FE,EA,AB,BH,HF)",
"Collinear(FHB)"
] | [
"Equal(LengthOfLine(AE),6)",
"Equal(LengthOfLine(DH),6)",
"Equal(LengthOfLine(EF),6)",
"PerpendicularBetweenLine(AB,FB)",
"PerpendicularBetweenLine(DH,BH)",
"PerpendicularBetweenLine(EA,BA)",
"PerpendicularBetweenLine(FE,AE)"
] | [
"Equal(LengthOfLine(AE),6)",
"Equal(LengthOfLine(DH),6)",
"Equal(LengthOfLine(EF),6)",
"PerpendicularBetweenLine(AB,FB)",
"PerpendicularBetweenLine(DH,BH)",
"PerpendicularBetweenLine(EA,BA)",
"PerpendicularBetweenLine(FE,AE)"
] | Value(Add(AreaOfTriangle(DFB),AreaOfQuadrilateral(FEAB))) | 54 | [
"parallel_judgment_ipsilateral_internal_angle(1,EF,AB)",
"parallel_judgment_ipsilateral_internal_angle(1,AE,BF)",
"parallelogram_judgment_parallel_and_parallel(1,FEAB)",
"parallelogram_property_opposite_line_equal(1,EABF)",
"parallelogram_area_formula_sine(1,FEAB)",
"adjacent_complementary_angle(1,FHD,DHB... | {"START": ["parallel_judgment_ipsilateral_internal_angle(1,EF,AB)", "parallel_judgment_ipsilateral_internal_angle(1,AE,BF)", "adjacent_complementary_angle(1,FHD,DHB)", "triangle_area_formula_common(1,DFB)"], "adjacent_complementary_angle(1,FHD,DHB)": ["altitude_of_triangle_judgment(1,DH,DFB)"], "parallel_judgment_ipsil... | |
162 | XiaokaiZhang_2023-04-02 | Geometry3k-164 | 4 | 如图所示，∠ANE=30°，∠CEB=110°，∠ECA=∠AFN，∠ENJ=130°。求∠CAE的大小。 | As shown in the diagram, ∠ANE=30°, ∠CEB=110°, ∠ECA=∠AFN, ∠ENJ=130°. Find the measure of ∠CAE. | 162.png | [
"Shape(CA,AE,EC)",
"Shape(EA,AN,NE)",
"Shape(AF,FN,NA)",
"Shape(CE,EB)",
"Shape(BE,EN)",
"Shape(EN,NJ)",
"Collinear(CAF)",
"Collinear(CEN)",
"Collinear(AEB)",
"Collinear(FNJ)"
] | [
"Equal(MeasureOfAngle(ANE),30)",
"Equal(MeasureOfAngle(CEB),110)",
"Equal(MeasureOfAngle(ECA),MeasureOfAngle(AFN))",
"Equal(MeasureOfAngle(ENJ),130)"
] | [
"Equal(MeasureOfAngle(ANE),30)",
"Equal(MeasureOfAngle(CEB),110)",
"Equal(MeasureOfAngle(ECA),MeasureOfAngle(AFN))",
"Equal(MeasureOfAngle(ENJ),130)"
] | Value(MeasureOfAngle(CAE)) | 45 | [
"adjacent_complementary_angle(1,AEC,CEB)",
"adjacent_complementary_angle(1,FNE,ENJ)",
"triangle_property_angle_sum(1,CFN)",
"triangle_property_angle_sum(1,CAE)"
] | {"START": ["adjacent_complementary_angle(1,AEC,CEB)", "adjacent_complementary_angle(1,FNE,ENJ)", "triangle_property_angle_sum(1,CFN)", "triangle_property_angle_sum(1,CAE)"]} | |
163 | XiaokaiZhang_2023-03-12 | Geometry3k-165 | 3 | 如图所示，KJ=11，KL=11，ML=5.5，∠KJM=60°，KM垂直于LM。求直线JM的长度。 | As shown in the diagram, KJ=11, KL=11, ML=5.5, ∠KJM=60°, KM is perpendicular to LM. Find the length of line JM. | 163.png | [
"Shape(KJ,JM,MK)",
"Shape(KM,ML,LK)",
"Collinear(JML)"
] | [
"Equal(LengthOfLine(KJ),11)",
"Equal(LengthOfLine(KL),11)",
"Equal(LengthOfLine(ML),5.5)",
"Equal(MeasureOfAngle(KJM),60)",
"PerpendicularBetweenLine(KM,LM)"
] | [
"Equal(LengthOfLine(KJ),11)",
"Equal(LengthOfLine(KL),11)",
"Equal(LengthOfLine(ML),5.5)",
"Equal(MeasureOfAngle(KJM),60)",
"PerpendicularBetweenLine(KM,LM)"
] | Value(LengthOfLine(JM)) | 11/2 | [
"adjacent_complementary_angle(1,JMK,KML)",
"triangle_property_angle_sum(1,JMK)",
"sine_theorem(1,JMK)"
] | {"START": ["adjacent_complementary_angle(1,JMK,KML)", "triangle_property_angle_sum(1,JMK)", "sine_theorem(1,JMK)"]} | |
164 | XiaokaiZhang_2023-03-12 | Geometry3k-166 | 2 | 如图所示，∠CAD=42°，∠CFG=77°，三角形CDE为等边三角形，CA和CB是等腰三角形CAB的腰，△CFG为等腰△。求∠GCF的大小。 | As shown in the diagram, ∠CAD=42°, ∠CFG=77°, △CDE is an equilateral △, △CAB is an isosceles △, triangleCFG is an isosceles triangle. Find the measure of ∠GCF. | 164.png | [
"Shape(CA,AD,DC)",
"Shape(CD,DF,FC)",
"Shape(CF,FG,GC)",
"Shape(CG,GE,EC)",
"Shape(CE,EB,BC)",
"Collinear(ADFGEB)"
] | [
"Equal(MeasureOfAngle(CAD),42)",
"Equal(MeasureOfAngle(CFG),77)",
"EquilateralTriangle(CDE)",
"IsoscelesTriangle(CAB)",
"IsoscelesTriangle(CFG)"
] | [
"Equal(MeasureOfAngle(CAD),42)",
"Equal(MeasureOfAngle(CFG),77)"
] | Value(MeasureOfAngle(GCF)) | 26 | [
"triangle_property_angle_sum(1,CFG)",
"isosceles_triangle_property_angle_equal(1,CFG)"
] | {"START": ["triangle_property_angle_sum(1,CFG)", "isosceles_triangle_property_angle_equal(1,CFG)"]} | |
165 | XiaokaiZhang_2023-04-02 | Geometry3k-167 | 3 | 如图所示，∠LWX=53°，WL平行于XE，XN平行于ZK。求∠XZK的大小。 | As shown in the diagram, ∠LWX=53°, WL is parallel to XE, XN∥ZK. Find the measure of ∠XZK. | 165.png | [
"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(EX... | [
"Equal(MeasureOfAngle(LWX),53)",
"ParallelBetweenLine(WL,XE)",
"ParallelBetweenLine(XN,ZK)"
] | [
"ParallelBetweenLine(WL,XE)",
"ParallelBetweenLine(XN,ZK)"
] | Value(MeasureOfAngle(XZK)) | 53 | [
"parallel_property_ipsilateral_internal_angle(1,WL,XE)",
"vertical_angle(1,WXE,NXZ)",
"parallel_property_ipsilateral_internal_angle(1,XN,ZK)"
] | {"START": ["parallel_property_ipsilateral_internal_angle(1,WL,XE)", "vertical_angle(1,WXE,NXZ)", "parallel_property_ipsilateral_internal_angle(1,XN,ZK)"]} | |
166 | XiaokaiZhang_2023-04-02 | Geometry3k-168 | 1 | 如图所示，∠MRQ=5*x°，∠PMR=x°，∠QPM=4*x°，∠RQP=2*x°。求∠PMR的大小。 | As shown in the diagram, ∠MRQ=5*x°, ∠PMR=x°, ∠QPM=4*x°, ∠RQP=2*x°. Find the measure of ∠PMR. | 166.png | [
"Shape(MR,RQ,QP,PM)"
] | [
"Equal(MeasureOfAngle(MRQ),5*x)",
"Equal(MeasureOfAngle(PMR),x)",
"Equal(MeasureOfAngle(QPM),4*x)",
"Equal(MeasureOfAngle(RQP),2*x)"
] | [
"Equal(MeasureOfAngle(MRQ),5*x)",
"Equal(MeasureOfAngle(PMR),x)",
"Equal(MeasureOfAngle(QPM),4*x)",
"Equal(MeasureOfAngle(RQP),2*x)"
] | Value(MeasureOfAngle(PMR)) | 30 | [
"quadrilateral_property_angle_sum(1,MRQP)"
] | {"START": ["quadrilateral_property_angle_sum(1,MRQP)"]} | |
167 | XiaokaiZhang_2023-03-12 | Geometry3k-169 | 1 | 如图所示，AB=15，BC=h，∠CAB=45°，BC垂直于AC。求h的值。 | As shown in the diagram, AB=15, BC=h, ∠CAB=45°, BC is perpendicular to AC. Find the value of h. | 167.png | [
"Shape(CA,AB,BC)"
] | [
"Equal(LengthOfLine(AB),15)",
"Equal(LengthOfLine(BC),h)",
"Equal(MeasureOfAngle(CAB),45)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),15)",
"Equal(LengthOfLine(BC),h)",
"Equal(MeasureOfAngle(CAB),45)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(h) | 15*sqrt(2)/2 | [
"sine_theorem(1,BCA)"
] | {"START": ["sine_theorem(1,BCA)"]} | |
168 | XiaokaiZhang_2023-03-12 | Geometry3k-170 | 2 | 如图所示，AC=18，AD=24，BC=x，BD=9，∠BAC=∠DAB。求x的值。 | As shown in the diagram, AC=18, AD=24, BC=x, BD=9, ∠BAC=∠DAB. Find the value of x. | 168.png | [
"Shape(CB,BA,AC)",
"Shape(BD,DA,AB)",
"Collinear(CBD)"
] | [
"Equal(LengthOfLine(AC),18)",
"Equal(LengthOfLine(AD),24)",
"Equal(LengthOfLine(BC),x)",
"Equal(LengthOfLine(BD),9)",
"Equal(MeasureOfAngle(BAC),MeasureOfAngle(DAB))"
] | [
"Equal(LengthOfLine(AC),18)",
"Equal(LengthOfLine(AD),24)",
"Equal(LengthOfLine(BC),x)",
"Equal(LengthOfLine(BD),9)",
"Equal(MeasureOfAngle(BAC),MeasureOfAngle(DAB))"
] | Value(x) | 27/4 | [
"bisector_of_angle_judgment_angle_equal(1,AB,DAC)",
"bisector_of_angle_property_line_ratio(1,AB,DAC)"
] | {"START": ["bisector_of_angle_judgment_angle_equal(1,AB,DAC)"], "bisector_of_angle_judgment_angle_equal(1,AB,DAC)": ["bisector_of_angle_property_line_ratio(1,AB,DAC)"]} | |
169 | XiaokaiZhang_2023-04-02 | Geometry3k-171 | 1 | 如图所示，AB=2*x+3，BC=5*x，∠CBA=80°，ADCB是菱形。求∠DCB的大小。 | As shown in the diagram, AB=2*x+3, BC=5*x, ∠CBA=80°, quadrilateral ADCB is a rhombus. Find the measure of ∠DCB. | 169.png | [
"Shape(AD,DE,EA)",
"Shape(ED,DC,CE)",
"Shape(EC,CB,BE)",
"Shape(AE,EB,BA)",
"Collinear(AEC)",
"Collinear(DEB)"
] | [
"Equal(LengthOfLine(AB),2*x+3)",
"Equal(LengthOfLine(BC),5*x)",
"Equal(MeasureOfAngle(CBA),80)",
"Rhombus(ADCB)"
] | [
"Equal(LengthOfLine(AB),2*x+3)",
"Equal(LengthOfLine(BC),5*x)",
"Equal(MeasureOfAngle(CBA),80)"
] | Value(MeasureOfAngle(DCB)) | 100 | [
"parallel_property_ipsilateral_internal_angle(1,CD,BA)"
] | {"START": ["parallel_property_ipsilateral_internal_angle(1,CD,BA)"]} | |
170 | XiaokaiZhang_2023-04-02 | Geometry3k-172 | 1 | 如图所示，∠PSR=x+10°，∠QPS=x°，∠RQP=2*x-16°，∠SRQ=2*x°。求∠SRQ的大小。 | As shown in the diagram, ∠PSR=x+10°, ∠QPS=x°, ∠RQP=2*x-16°, ∠SRQ=2*x°. Find the measure of ∠SRQ. | 170.png | [
"Shape(QP,PS,SR,RQ)"
] | [
"Equal(MeasureOfAngle(PSR),x+10)",
"Equal(MeasureOfAngle(QPS),x)",
"Equal(MeasureOfAngle(RQP),2*x-16)",
"Equal(MeasureOfAngle(SRQ),2*x)"
] | [
"Equal(MeasureOfAngle(PSR),x+10)",
"Equal(MeasureOfAngle(QPS),x)",
"Equal(MeasureOfAngle(RQP),2*x-16)",
"Equal(MeasureOfAngle(SRQ),2*x)"
] | Value(MeasureOfAngle(SRQ)) | 122 | [
"quadrilateral_property_angle_sum(1,QPSR)"
] | {"START": ["quadrilateral_property_angle_sum(1,QPSR)"]} | |
171 | XiaokaiZhang_2023-03-12 | Geometry3k-173 | 2 | 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。 | 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. | 171.png | [
"Shape(JL,LK,KJ)",
"Shape(KL,LN,NK)",
"Shape(NL,LM,MN)"
] | [
"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)"
] | [
"Equal(LengthOfLine(KL),LengthOfLine(NL))",
"Equal(LengthOfLine(NM),LengthOfLine(ML))",
"Equal(MeasureOfAngle(JLK),25)",
"Equal(MeasureOfAngle(KLN),18)",
"Equal(MeasureOfAngle(NLM),20)"
] | Value(MeasureOfAngle(LMN)) | 140 | [
"isosceles_triangle_property_angle_equal(1,MNL)",
"triangle_property_angle_sum(1,NLM)"
] | {"START": ["isosceles_triangle_property_angle_equal(1,MNL)", "triangle_property_angle_sum(1,NLM)"]} | |
172 | XiaokaiZhang_2023-04-02 | Geometry3k-174 | 4 | 如图所示，∠AGC=60°，DG⊥AG。求∠EGD的大小。 | As shown in the diagram, ∠AGC=60°, DG is perpendicular to AG. Find the measure of ∠EGD. | 172.png | [
"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)"
] | [
"Equal(MeasureOfAngle(AGC),60)",
"PerpendicularBetweenLine(DG,AG)"
] | [
"Equal(MeasureOfAngle(AGC),60)",
"PerpendicularBetweenLine(DG,AG)"
] | Value(MeasureOfAngle(EGD)) | 30 | [
"adjacent_complementary_angle(1,AGC,CGB)",
"adjacent_complementary_angle(1,BGD,DGA)",
"adjacent_complementary_angle(1,CGB,BGE)",
"angle_addition(1,BGE,EGD)"
] | {"START": ["adjacent_complementary_angle(1,AGC,CGB)", "adjacent_complementary_angle(1,BGD,DGA)", "adjacent_complementary_angle(1,CGB,BGE)", "angle_addition(1,BGE,EGD)"]} | |
173 | XiaokaiZhang_2023-04-02 | Geometry3k-175 | 11 | 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。 | As shown in the diagram, QR=2, QW=15, ST=5, XW=12, WS∥VT, XR∥WS. Find the length of line WV. | 173.png | [
"Shape(QX,XR,RQ)",
"Shape(XW,WS,SR,RX)",
"Shape(WV,VT,TS,SW)",
"Collinear(QXWV)",
"Collinear(QRST)"
] | [
"Equal(LengthOfLine(QR),2)",
"Equal(LengthOfLine(QW),15)",
"Equal(LengthOfLine(ST),5)",
"Equal(LengthOfLine(XW),12)",
"ParallelBetweenLine(WS,VT)",
"ParallelBetweenLine(XR,WS)"
] | [
"ParallelBetweenLine(WS,VT)",
"ParallelBetweenLine(XR,WS)"
] | Value(LengthOfLine(WV)) | 15/2 | [
"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... | {"START": ["parallel_property_corresponding_angle(1,XR,WS,Q)", "parallel_property_corresponding_angle(1,WS,VT,X)", "line_addition(1,QX,XW)", "line_addition(1,QS,ST)", "line_addition(1,QW,WV)"], "parallel_property_corresponding_angle(1,WS,VT,X)": ["similar_triangle_judgment_aa(1,SQW,TQV)"], "parallel_property_correspond... | |
174 | XiaokaiZhang_2023-04-02 | Geometry3k-176 | 2 | 如图所示，∠EAD=42°，圆A的圆心为A。求弧AEC的角度。 | As shown in the diagram, ∠EAD=42°, A is the center of circle A. Find the measure of ⌒AEC. | 174.png | [
"Shape(AE,AEB,BA)",
"Shape(AB,ABC,CA)",
"Shape(AC,ACD,DA)",
"Shape(AD,ADE,EA)",
"Collinear(BAD)",
"Collinear(CAE)",
"Cocircular(A,BCDE)"
] | [
"Equal(MeasureOfAngle(EAD),42)",
"IsCentreOfCircle(A,A)"
] | [
"Equal(MeasureOfAngle(EAD),42)",
"IsCentreOfCircle(A,A)"
] | Value(MeasureOfArc(AEC)) | 180 | [
"flat_angle(1,CAE)",
"arc_property_center_angle(1,AEC,A)"
] | {"START": ["flat_angle(1,CAE)", "arc_property_center_angle(1,AEC,A)"]} | |
175 | XiaokaiZhang_2023-04-02 | Geometry3k-177 | 2 | 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。 | 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. | 175.png | [
"Shape(BA,CGA,GB)",
"Shape(AC,CG,CGA)",
"Shape(CA,CAG,GC)",
"Collinear(CGB)",
"Cocircular(C,AG)"
] | [
"Equal(LengthOfLine(CA),15)",
"Equal(LengthOfLine(CB),x)",
"Equal(MeasureOfAngle(GBA),30)",
"IsCentreOfCircle(C,C)",
"IsTangentOfCircle(BA,C)"
] | [
"Equal(LengthOfLine(CA),15)",
"Equal(LengthOfLine(CB),x)",
"Equal(MeasureOfAngle(GBA),30)",
"IsCentreOfCircle(C,C)"
] | Value(x) | 30 | [
"tangent_of_circle_property_perpendicular(2,BA,C,C)",
"sine_theorem(1,CBA)"
] | {"START": ["tangent_of_circle_property_perpendicular(2,BA,C,C)", "sine_theorem(1,CBA)"]} | |
176 | XiaokaiZhang_2023-03-12 | Geometry3k-178 | 4 | 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。 | 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. | 176.png | [
"Shape(CA,AD,DC)",
"Shape(CD,DF,FC)",
"Shape(CF,FG,GC)",
"Shape(CG,GE,EC)",
"Shape(CE,EB,BC)",
"Collinear(ADFGEB)"
] | [
"Equal(MeasureOfAngle(CAD),42)",
"Equal(MeasureOfAngle(CFG),77)",
"EquilateralTriangle(CDE)",
"IsoscelesTriangle(CAB)",
"IsoscelesTriangle(CFG)"
] | [
"Equal(MeasureOfAngle(CAD),42)",
"Equal(MeasureOfAngle(CFG),77)"
] | Value(MeasureOfAngle(FCD)) | 17 | [
"angle_addition(1,DFC,CFG)",
"flat_angle(1,AFG)",
"equilateral_triangle_property_angle(1,DEC)",
"triangle_property_angle_sum(1,CDF)"
] | {"START": ["angle_addition(1,DFC,CFG)", "flat_angle(1,AFG)", "equilateral_triangle_property_angle(1,DEC)", "triangle_property_angle_sum(1,CDF)"]} | |
177 | XiaokaiZhang_2023-03-12 | Geometry3k-179 | 3 | 如图所示，AB=y，AC=5，BC=x，∠BAC=60°，AC垂直于BC。求x的值。 | As shown in the diagram, AB=y, AC=5, BC=x, ∠BAC=60°, AC⊥BC. Find the value of x. | 177.png | [
"Shape(AC,CB,BA)"
] | [
"Equal(LengthOfLine(AB),y)",
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(BAC),60)",
"PerpendicularBetweenLine(AC,BC)"
] | [
"Equal(LengthOfLine(AB),y)",
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(BAC),60)",
"PerpendicularBetweenLine(AC,BC)"
] | Value(x) | 5*sqrt(3) | [
"triangle_property_angle_sum(1,ACB)",
"sine_theorem(1,ACB)",
"sine_theorem(1,BAC)"
] | {"START": ["triangle_property_angle_sum(1,ACB)", "sine_theorem(1,ACB)", "sine_theorem(1,BAC)"]} | |
178 | XiaokaiZhang_2023-04-02 | Geometry3k-180 | 1 | 如图所示，CB=18，CD=12，∠BAD=115°，四边形ADCB是▱。求∠ADC的大小。 | 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. | 178.png | [
"Shape(AD,DC,CB,BA)"
] | [
"Equal(LengthOfLine(CB),18)",
"Equal(LengthOfLine(CD),12)",
"Equal(MeasureOfAngle(BAD),115)",
"Parallelogram(ADCB)"
] | [
"Equal(LengthOfLine(CB),18)",
"Equal(LengthOfLine(CD),12)",
"Equal(MeasureOfAngle(BAD),115)"
] | Value(MeasureOfAngle(ADC)) | 65 | [
"parallel_property_ipsilateral_internal_angle(1,AB,DC)"
] | {"START": ["parallel_property_ipsilateral_internal_angle(1,AB,DC)"]} | |
179 | XiaokaiZhang_2023-04-02 | Geometry3k-181 | 1 | 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。 | 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. | 179.png | [
"Shape(QX,XY,YR,RQ)",
"Shape(XT,TS,SY,YX)",
"Collinear(QXT)",
"Collinear(RYS)"
] | [
"Equal(LengthOfLine(QR),22)",
"Equal(LengthOfLine(XY),15)",
"IsMidsegmentOfQuadrilateral(XY,QTSR)",
"IsoscelesTrapezoid(QTSR)"
] | [] | Value(LengthOfLine(TS)) | 8 | [
"midsegment_of_quadrilateral_property_length(1,XY,QTSR)"
] | {"START": ["midsegment_of_quadrilateral_property_length(1,XY,QTSR)"]} | |
180 | XiaokaiZhang_2023-04-02 | Geometry3k-182 | 1 | 如图所示，∠FHM=94°，HM∥CA。求∠HCA的大小。 | As shown in the diagram, ∠FHM=94°, HM∥CA. Find the measure of ∠HCA. | 180.png | [
"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)"
] | [
"Equal(MeasureOfAngle(FHM),94)",
"ParallelBetweenLine(HM,CA)"
] | [
"Equal(MeasureOfAngle(FHM),94)",
"ParallelBetweenLine(HM,CA)"
] | Value(MeasureOfAngle(HCA)) | 94 | [
"parallel_property_corresponding_angle(1,HM,CA,F)"
] | {"START": ["parallel_property_corresponding_angle(1,HM,CA,F)"]} | |
181 | XiaokaiZhang_2023-04-02 | Geometry3k-183 | 2 | 如图所示，∠YZW=60°，∠ZWX=95°。求∠XYZ的大小。 | As shown in the diagram, ∠YZW=60°, ∠ZWX=95°. Find the measure of ∠XYZ. | 181.png | [
"Shape(VZW,WZ)",
"Shape(VWX,XW)",
"Shape(WXY,YX)",
"Shape(VYZ,ZY)",
"Shape(ZW,WX,XY,YZ)",
"Cocircular(V,ZWXY)"
] | [
"Equal(MeasureOfAngle(YZW),60)",
"Equal(MeasureOfAngle(ZWX),95)"
] | [
"Equal(MeasureOfAngle(YZW),60)",
"Equal(MeasureOfAngle(ZWX),95)"
] | Value(MeasureOfAngle(XYZ)) | 85 | [
"arc_property_circumference_angle_external(1,VZX,Y)",
"arc_property_circumference_angle_internal(1,VZX,W)"
] | {"START": ["arc_property_circumference_angle_external(1,VZX,Y)", "arc_property_circumference_angle_internal(1,VZX,W)"]} | |
182 | XiaokaiZhang_2023-04-02 | Geometry3k-184 | 1 | 如图所示，∠ADG=36°，∠AGF=104°，∠EFC=40°，GB⊥CB。求∠DGA的大小。 | As shown in the diagram, ∠ADG=36°, ∠AGF=104°, ∠EFC=40°, GB⊥CB. Find the measure of ∠DGA. | 182.png | [
"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)"
] | [
"Equal(MeasureOfAngle(ADG),36)",
"Equal(MeasureOfAngle(AGF),104)",
"Equal(MeasureOfAngle(EFC),40)",
"PerpendicularBetweenLine(GB,CB)"
] | [
"Equal(MeasureOfAngle(ADG),36)",
"Equal(MeasureOfAngle(AGF),104)",
"Equal(MeasureOfAngle(EFC),40)"
] | Value(MeasureOfAngle(DGA)) | 76 | [
"adjacent_complementary_angle(1,DGA,AGF)"
] | {"START": ["adjacent_complementary_angle(1,DGA,AGF)"]} | |
183 | XiaokaiZhang_2023-04-02 | Geometry3k-185 | 1 | 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。 | As shown in the diagram, the area of XWZY is 100, XZ=10, XWZY is a rhombus. Find the length of line WY. | 183.png | [
"Shape(XW,WA,AX)",
"Shape(AW,WZ,ZA)",
"Shape(AZ,ZY,YA)",
"Shape(XA,AY,YX)",
"Collinear(WAY)",
"Collinear(XAZ)"
] | [
"Equal(AreaOfQuadrilateral(XWZY),100)",
"Equal(LengthOfLine(XZ),10)",
"Rhombus(XWZY)"
] | [] | Value(LengthOfLine(WY)) | 20 | [
"kite_area_formula_diagonal(1,XWZY)"
] | {"START": ["kite_area_formula_diagonal(1,XWZY)"]} | |
184 | XiaokaiZhang_2023-04-02 | Geometry3k-186 | 2 | 如图所示，CB∥DE。求证∠TCB+∠EDG=180°。 | As shown in the diagram, CB∥DE. Prove that ∠TCB+∠EDG=180°. | 184.png | [
"Shape(TC,CB)",
"Shape(CD,DE,EB,BC)",
"Shape(ED,DG)",
"Collinear(TCDG)"
] | [
"ParallelBetweenLine(CB,DE)"
] | [] | Equal(Add(MeasureOfAngle(TCB),MeasureOfAngle(EDG)),180) | 0 | [
"parallel_property_corresponding_angle(1,CB,DE,T)",
"adjacent_complementary_angle(1,CDE,EDG)"
] | {"START": ["parallel_property_corresponding_angle(1,CB,DE,T)", "adjacent_complementary_angle(1,CDE,EDG)"]} | |
185 | XiaokaiZhang_2023-04-02 | Geometry3k-187 | 6 | 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。 | 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. | 185.png | [
"Shape(ACD,DA,AC)",
"Shape(AD,ADE,EA)",
"Shape(AE,AEB,BA)",
"Shape(AB,ABC,CA)",
"Collinear(CAE)",
"Collinear(DAB)",
"Cocircular(A,CDEB)"
] | [
"Equal(LengthOfLine(AC),3)",
"Equal(MeasureOfAngle(DAC),MeasureOfAngle(BAE))",
"Equal(MeasureOfAngle(EAD),130)",
"IsCentreOfCircle(A,A)"
] | [
"Equal(LengthOfLine(AC),3)",
"Equal(MeasureOfAngle(DAC),MeasureOfAngle(BAE))",
"Equal(MeasureOfAngle(EAD),130)",
"IsCentreOfCircle(A,A)"
] | Value(Add(AreaOfSector(ACD),AreaOfSector(AEB))) | 5*pi/2 | [
"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)"
] | {"START": ["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)"]} | |
186 | XiaokaiZhang_2023-04-02 | Geometry3k-188 | 1 | 如图所示，∠SRU=6*x-54°，∠UTS=4*x+6°，TSRU是平行四边形。求x的值。 | As shown in the diagram, ∠SRU=6*x-54°, ∠UTS=4*x+6°, quadrilateral TSRU is a parallelogram. Find the value of x. | 186.png | [
"Shape(TS,SR,RU,UT)"
] | [
"Equal(MeasureOfAngle(SRU),6*x-54)",
"Equal(MeasureOfAngle(UTS),4*x+6)",
"Parallelogram(TSRU)"
] | [
"Equal(MeasureOfAngle(SRU),6*x-54)",
"Equal(MeasureOfAngle(UTS),4*x+6)"
] | Value(x) | 30 | [
"parallelogram_property_opposite_angle_equal(1,TSRU)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,TSRU)"]} | |
187 | XiaokaiZhang_2023-03-12 | Geometry3k-189 | 0 | 如图所示，AC=27，BA=2*x+5，BA=BC，BC=3*x-4。求直线BC的长度。 | As shown in the diagram, AC=27, BA=2*x+5, BA=BC, BC=3*x-4. Find the length of line BC. | 187.png | [
"Shape(AB,BC,CA)"
] | [
"Equal(LengthOfLine(AC),27)",
"Equal(LengthOfLine(BA),2*x+5)",
"Equal(LengthOfLine(BA),LengthOfLine(BC))",
"Equal(LengthOfLine(BC),3*x-4)"
] | [
"Equal(LengthOfLine(AC),27)",
"Equal(LengthOfLine(BA),2*x+5)",
"Equal(LengthOfLine(BA),LengthOfLine(BC))",
"Equal(LengthOfLine(BC),3*x-4)"
] | Value(LengthOfLine(BC)) | 23 | [] | {"START": []} | |
188 | XiaokaiZhang_2023-04-02 | Geometry3k-190 | 10 | 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。 | 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. | 188.png | [
"Shape(BD,BDE,EB)",
"Shape(BE,BED,DB)",
"Collinear(CDBEA)",
"Cocircular(C,D)",
"Cocircular(B,DE)",
"Cocircular(A,E)"
] | [
"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)"
] | [
"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)"
] | Value(LengthOfLine(AC)) | 27 | [
"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)",
"li... | {"START": ["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_addi... | |
189 | XiaokaiZhang_2023-03-12 | Geometry3k-191 | 4 | 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。 | 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. | 189.png | [
"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)"
] | [
"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)"
] | [
"Equal(LengthOfLine(AE),12)",
"Equal(LengthOfLine(AJ),15)",
"Equal(MeasureOfAngle(JBE),34)",
"Equal(MeasureOfAngle(JCD),32)",
"PerpendicularBetweenLine(AF,JF)",
"PerpendicularBetweenLine(CD,JD)",
"PerpendicularBetweenLine(JE,AE)"
] | Value(MeasureOfAngle(JAC)) | 24 | [
"angle_addition(1,FCJ,JCD)",
"angle_addition(1,DBJ,JBE)",
"angle_addition(1,EAJ,JAF)",
"triangle_property_angle_sum(1,CBA)"
] | {"START": ["angle_addition(1,FCJ,JCD)", "angle_addition(1,DBJ,JBE)", "angle_addition(1,EAJ,JAF)", "triangle_property_angle_sum(1,CBA)"]} | |
190 | XiaokaiZhang_2023-04-02 | Geometry3k-192 | 1 | 如图所示，∠UVT=23°。求⌒DTU的角度。 | As shown in the diagram, ∠UVT=23°. Find the measure of ⌒DTU. | 190.png | [
"Shape(DVT,TV)",
"Shape(VT,DTU,UV)",
"Shape(VU,DUV)",
"Cocircular(D,VTU)"
] | [
"Equal(MeasureOfAngle(UVT),23)"
] | [
"Equal(MeasureOfAngle(UVT),23)"
] | Value(MeasureOfArc(DTU)) | 46 | [
"arc_property_circumference_angle_external(1,DTU,V)"
] | {"START": ["arc_property_circumference_angle_external(1,DTU,V)"]} | |
191 | XiaokaiZhang_2023-03-12 | Geometry3k-193 | 5 | 如图所示，VW=3*x-6，WX=x+4，YW=5，ZW=6，YX∥VZ。求直线WX的长度。 | As shown in the diagram, VW=3*x-6, WX=x+4, YW=5, ZW=6, YX∥VZ. Find the length of line WX. | 191.png | [
"Shape(VZ,ZW,WV)",
"Shape(WX,XY,YW)",
"Collinear(ZWY)",
"Collinear(VWX)"
] | [
"Equal(LengthOfLine(VW),3*x-6)",
"Equal(LengthOfLine(WX),x+4)",
"Equal(LengthOfLine(YW),5)",
"Equal(LengthOfLine(ZW),6)",
"ParallelBetweenLine(YX,VZ)"
] | [
"Equal(LengthOfLine(VW),3*x-6)",
"Equal(LengthOfLine(WX),x+4)",
"Equal(LengthOfLine(YW),5)",
"Equal(LengthOfLine(ZW),6)",
"ParallelBetweenLine(YX,VZ)"
] | Value(LengthOfLine(WX)) | 10 | [
"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)"
] | {"START": ["parallel_property_alternate_interior_angle(1,YX,VZ)", "parallel_property_alternate_interior_angle(2,YX,VZ)"], "parallel_property_alternate_interior_angle(1,YX,VZ)": ["similar_triangle_judgment_aa(1,WVZ,WXY)"], "parallel_property_alternate_interior_angle(2,YX,VZ)": ["similar_triangle_judgment_aa(1,WVZ,WXY)"]... | |
192 | XiaokaiZhang_2023-04-02 | Geometry3k-194 | 1 | 如图所示，∠SRU=23°，弧BVT的角度为68。求弧BSU的角度。 | As shown in the diagram, ∠SRU=23°, the measure of arc BVT is 68. Find the measure of arc BSU. | 192.png | [
"Shape(SR,RU,BSU)",
"Shape(ST,BTS)",
"Shape(BSU,UV,BVT,TS)",
"Shape(BUV,VU)",
"Collinear(RST)",
"Collinear(RUV)",
"Cocircular(B,SUVT)"
] | [
"Equal(MeasureOfAngle(SRU),23)",
"Equal(MeasureOfArc(BVT),68)"
] | [
"Equal(MeasureOfAngle(SRU),23)",
"Equal(MeasureOfArc(BVT),68)"
] | Value(MeasureOfArc(BSU)) | 22 | [
"circle_property_circular_power_segment_and_segment_angle(1,RST,RUV,B)"
] | {"START": ["circle_property_circular_power_segment_and_segment_angle(1,RST,RUV,B)"]} | |
193 | XiaokaiZhang_2023-03-12 | Geometry3k-195 | 4 | 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。 | As shown in the diagram, AB=12, AC=y, AD=4, BC=x, CD=z, BA⊥CA, DC⊥BC. Find the value of x. | 193.png | [
"Shape(CB,BA,AC)",
"Shape(CA,AD,DC)",
"Collinear(BAD)"
] | [
"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)"
] | [
"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)"
] | Value(x) | 8*sqrt(3) | [
"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)"
] | {"START": ["line_addition(1,BA,AD)", "mirror_similar_triangle_judgment_aa(1,CBA,DCB)"], "mirror_similar_triangle_judgment_aa(1,CBA,DCB)": ["mirror_similar_triangle_property_line_ratio(1,CBA,DCB)", "mirror_similar_triangle_property_line_ratio(1,ACB,CBD)"]} | |
194 | XiaokaiZhang_2023-04-02 | Geometry3k-196 | 1 | 如图所示，AE=5，AF=x，AH=6，GA=12。求x的值。 | As shown in the diagram, AE=5, AF=x, AH=6, GA=12. Find the value of x. | 194.png | [
"Shape(AG,JGF,FA)",
"Shape(AF,JFE,EA)",
"Shape(AE,JEH,HA)",
"Shape(AH,JHG,GA)",
"Collinear(FAH)",
"Collinear(GAE)",
"Cocircular(J,FEHG)"
] | [
"Equal(LengthOfLine(AE),5)",
"Equal(LengthOfLine(AF),x)",
"Equal(LengthOfLine(AH),6)",
"Equal(LengthOfLine(GA),12)"
] | [
"Equal(LengthOfLine(AE),5)",
"Equal(LengthOfLine(AF),x)",
"Equal(LengthOfLine(AH),6)",
"Equal(LengthOfLine(GA),12)"
] | Value(x) | 10 | [
"circle_property_circular_power_chord_and_chord(1,FAH,GAE,J)"
] | {"START": ["circle_property_circular_power_chord_and_chord(1,FAH,GAE,J)"]} | |
195 | XiaokaiZhang_2023-04-02 | Geometry3k-197 | 4 | 如图所示，∠LWX=53°，WL∥XE，XN∥ZK。求∠ZYM的大小。 | As shown in the diagram, ∠LWX=53°, WL is parallel to XE, XN∥ZK. Find the measure of ∠ZYM. | 195.png | [
"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(HZ... | [
"Equal(MeasureOfAngle(LWX),53)",
"ParallelBetweenLine(WL,XE)",
"ParallelBetweenLine(XN,ZK)"
] | [
"Equal(MeasureOfAngle(LWX),53)",
"ParallelBetweenLine(WL,XE)",
"ParallelBetweenLine(XN,ZK)"
] | Value(MeasureOfAngle(ZYM)) | 127 | [
"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)"
] | {"START": ["parallel_property_collinear_extend(1,XN,ZK,W)", "adjacent_complementary_angle(1,LWX,XWY)"], "parallel_property_collinear_extend(1,XN,ZK,W)": ["parallel_property_collinear_extend(2,KZ,XW,Y)"], "parallel_property_collinear_extend(2,KZ,XW,Y)": ["parallel_property_corresponding_angle(2,WX,YZ,M)"]} | |
196 | XiaokaiZhang_2023-03-12 | Geometry3k-198 | 1 | 如图所示，AC=11，BC=x，∠ABC=30°，∠CAB=120°。求x的值。 | As shown in the diagram, AC=11, BC=x, ∠ABC=30°, ∠CAB=120°. Find the value of x. | 196.png | [
"Shape(CA,AB,BC)"
] | [
"Equal(LengthOfLine(AC),11)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(ABC),30)",
"Equal(MeasureOfAngle(CAB),120)"
] | [
"Equal(LengthOfLine(AC),11)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(ABC),30)",
"Equal(MeasureOfAngle(CAB),120)"
] | Value(x) | 11*sqrt(3) | [
"sine_theorem(1,CAB)"
] | {"START": ["sine_theorem(1,CAB)"]} | |
197 | XiaokaiZhang_2023-04-02 | Geometry3k-199 | 1 | 如图所示，∠TWV=3*x-4°，∠UTW=x°，∠VUT=3*x-4°，∠WVU=x°。求∠UTW的大小。 | As shown in the diagram, ∠TWV=3*x-4°, ∠UTW=x°, ∠VUT=3*x-4°, ∠WVU=x°. Find the measure of ∠UTW. | 197.png | [
"Shape(UT,TW,WV,VU)"
] | [
"Equal(MeasureOfAngle(TWV),3*x-4)",
"Equal(MeasureOfAngle(UTW),x)",
"Equal(MeasureOfAngle(VUT),3*x-4)",
"Equal(MeasureOfAngle(WVU),x)"
] | [
"Equal(MeasureOfAngle(TWV),3*x-4)",
"Equal(MeasureOfAngle(UTW),x)",
"Equal(MeasureOfAngle(VUT),3*x-4)",
"Equal(MeasureOfAngle(WVU),x)"
] | Value(MeasureOfAngle(UTW)) | 46 | [
"quadrilateral_property_angle_sum(1,UTWV)"
] | {"START": ["quadrilateral_property_angle_sum(1,UTWV)"]} | |
198 | XiaokaiZhang_2023-04-02 | Geometry3k-200 | 1 | 如图所示，∠ABD=x+14°，∠ABF=3*x-8°，BD平分∠ABF。求∠ABD的大小。 | As shown in the diagram, ∠ABD=x+14°, ∠ABF=3*x-8°, BD bisects ∠ABF. Find the measure of ∠ABD. | 198.png | [
"Shape(AB,BD)",
"Shape(DB,BF)",
"Shape(FB,BC)",
"Collinear(ABC)"
] | [
"Equal(MeasureOfAngle(ABD),x+14)",
"Equal(MeasureOfAngle(ABF),3*x-8)",
"IsBisectorOfAngle(BD,ABF)"
] | [
"Equal(MeasureOfAngle(ABD),x+14)",
"Equal(MeasureOfAngle(ABF),3*x-8)",
"IsBisectorOfAngle(BD,ABF)"
] | Value(MeasureOfAngle(ABD)) | 50 | [
"angle_addition(1,ABD,DBF)"
] | {"START": ["angle_addition(1,ABD,DBF)"]} | |
199 | XiaokaiZhang_2023-04-02 | Geometry3k-201 | 1 | 如图所示，AJ=x，AK=x+2，AM=x+7，LA=x+10。求x的值。 | As shown in the diagram, AJ=x, AK=x+2, AM=x+7, LA=x+10. Find the value of x. | 199.png | [
"Shape(AJ,OJM,MA)",
"Shape(AM,OML,LA)",
"Shape(AL,OLK,KA)",
"Shape(AK,OKJ,JA)",
"Collinear(JAL)",
"Collinear(KAM)",
"Cocircular(O,MLKJ)"
] | [
"Equal(LengthOfLine(AJ),x)",
"Equal(LengthOfLine(AK),x+2)",
"Equal(LengthOfLine(AM),x+7)",
"Equal(LengthOfLine(LA),x+10)"
] | [
"Equal(LengthOfLine(AJ),x)",
"Equal(LengthOfLine(AK),x+2)",
"Equal(LengthOfLine(AM),x+7)",
"Equal(LengthOfLine(LA),x+10)"
] | Value(x) | 14 | [
"circle_property_circular_power_chord_and_chord(1,JAL,KAM,O)"
] | {"START": ["circle_property_circular_power_chord_and_chord(1,JAL,KAM,O)"]} | |
200 | XiaokaiZhang_2023-03-12 | Geometry3k-202 | 2 | 如图所示，CB=8，CF=x，FB=15，BC垂直于FC。求x的值。 | As shown in the diagram, CB=8, CF=x, FB=15, BC⊥FC. Find the value of x. | 200.png | [
"Shape(BC,CF,FB)"
] | [
"Equal(LengthOfLine(CB),8)",
"Equal(LengthOfLine(CF),x)",
"Equal(LengthOfLine(FB),15)",
"PerpendicularBetweenLine(BC,FC)"
] | [
"Equal(LengthOfLine(CB),8)",
"Equal(LengthOfLine(CF),x)",
"Equal(LengthOfLine(FB),15)",
"PerpendicularBetweenLine(BC,FC)"
] | Value(x) | sqrt(161) | [
"right_triangle_judgment_angle(1,BCF)",
"right_triangle_property_pythagorean(1,BCF)"
] | {"START": ["right_triangle_judgment_angle(1,BCF)"], "right_triangle_judgment_angle(1,BCF)": ["right_triangle_property_pythagorean(1,BCF)"]} |
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