2026年统编版适配八年级数学开学摸底卷空间想象与综合证明标准试卷第398套（含答案解析与可打印作答区）

八年级数学开学摸底卷空间想象与综合证明第398套请在规定位置作答，保持卷面整洁。2026年统编版适配八年级数学开学摸底卷空间想象与综合证明标准试卷第398套（含答案解析与可打印作答区）学校：________________班级：________姓名：________考号：________考试时间：120分钟满分：120分答题说明：本卷共26题。请先检查题号是否完整，再按要求在答题栏或作答区内作答；选择题只有一个最佳答案；解答题须写出必要的推理过程、计算步骤和结论。诚信提示：独立思考，规范表达；作图、计算和证明均以清楚、准确为准。题型选择题填空题基础解答题综合探究题总分分值30183636120得分一、选择题（本大题共10小题，每小题3分，共30分）请将正确选项填入下表。题号12345678910答案1.（3分）把六个相同的正方形按如下方式排成正方体展开图：1、2、3、4四个正方形从左到右排成一行，5在2的上方，6在3的下方。折成正方体后，与标号1的面相对的是（）A.2B.3C.5D.62.（3分）长方体的长、宽、高分别为6cm、4cm、3cm。一只蚂蚁从一个顶点沿表面爬到它的对顶点，若不重复绕行，最短路程为（）A.√85cmB.√97cmC.√109cmD.13cm3.（3分）在△ABC中，∠A=40°，∠B=60°，则∠C的外角中与∠A、∠B不相邻的那个外角等于（）A.80°B.100°C.120°D.140°4.（3分）已知△ABC与△DEF中，AB=DE，AC=DF，∠A=∠D。判定两个三角形全等可直接使用的依据是（）A.SSSB.SASC.ASAD.AAS5.（3分）两条平行直线被第三条直线所截，若其中一个内错角为72°，则与它同旁的内角为（）A.18°B.72°C.108°D.144°6.（3分）一个几何体由4个相同小正方体搭成：底层有3个小正方体排成“L”形，另1个小正方体叠在“L”形拐角小正方体的正上方。从正上方观察，看到的小正方形个数是（）A.2B.3C.4D.57.（3分）下列命题中，正确的是（）A.若两条直线都垂直于第三条直线，则这两条直线一定平行B.若两个角相等，则它们一定是对顶角C.若两条直线都平行于第三条直线，则这两条直线平行D.若两个三角形有两边相等，则这两个三角形全等

8.（3分）计算(x+2)(x-2)的结果是（）A.x²+4B.x²-4C.x²+2x-4D.x²-2x-49.（3分）不等式2x-3<5的解集是（）A.x<1B.x<4C.x>1D.x>410.（3分）若三角形两边长分别为5和7，第三边长为整数x，则x可以取的整数值共有（）A.8个B.9个C.10个D.11个二、填空题（本大题共6小题，每小题3分，共18分）11.（3分）长方体的长、宽、高分别为5cm、4cm、3cm，它的表面积是__________cm²。________________________________________________________________________________12.（3分）一个三角形的三个内角度数之比为2:3:4，则最大的内角为__________°。________________________________________________________________________________13.（3分）在正方体ABCD-A₁B₁C₁D₁中，直线AB与直线CC₁的位置关系是__________。________________________________________________________________________________14.（3分）若∠1与∠2互为补角，且∠1=3∠2-20°，则∠2=__________°。________________________________________________________________________________15.（3分）平面直角坐标系中，点A(-2,3)、B(4,3)，则线段AB的长为__________。________________________________________________________________________________16.（3分）棱长为2cm的正方体，它的体对角线长为__________cm。________________________________________________________________________________三、基础解答题（本大题共6小题，每小题6分，共36分）17.（6分）先化简，再求值：(x-3)²-(x+2)(x-2)，其中x=1/2。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（6分）如△ABC中，D是BC的中点，AD⊥BC。请证明AB=AC，并写出你使用的全等依据。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（6分）一个长方体木块长8cm、宽5cm、高4cm。从它的一角沿长、宽方向切去一个长3cm、宽2cm、高4cm的小长方体，得到一个直角形柱体。（1）求剩余几何体的体积；（2）说明为什么切去这块以后，上下底面的周长发生变化而侧面面积可用“底面周长×高”求得。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（6分）已知直线a∥b，一条截线分别与a、b相交。若在直线a处形成的一个锐角为35°，过直线b上的交点作该处钝角的角平分线，求角平分线与直线b所成的较小角，并说明理由。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（6分）把一张长方形纸片ABCD沿对角线AC折叠，使点B落到点B′处。若AB=6cm，BC=8cm。（1）求AC的长；（2）说明折叠后AB′与AB相等的理由；（3）若只看折叠前的长方形，判断△ABC的形状。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（6分）某立体由若干个同样的小正方体叠成。从正上方看，底面位置为三格一行，依次记为左、中、右；这三格上的小正方体高度分别为1、3、2。（1）求该立体共由多少个小正方体组成；（2）从正面看，可看到的每列最高层数分别是多少；（3）若每个小正方体棱长为1，求所有外露小正方形面的总数。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________四、综合探究题（本大题共4小题，每小题9分，共36分）23.（9分）在△ABC中，AB=AC，点D为BC的中点，连接AD。点E在AD上，且BE与CE分别连接。（1）证明AD⊥BC；（2）证明∠BED=∠CED；（3）若∠BAC=48°，求∠ABC的度数，并说明推理过程。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________24.（9分）如图形描述：长方体ABCD-A₁B₁C₁D₁中，AB=9cm，BC=6cm，AA₁=4cm。蚂蚁从A点沿长方体表面爬到C₁点。（1）列出三种不同展开方式对应的直线距离表达式；（2）求最短距离；（3）解释为什么“直接连接空间对角线”不能作为表面爬行路线。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________25.（9分）材料阅读与探究：观察凸多面体，若V表示顶点数，E表示棱数，F表示面数，许多凸多面体都满足V-E+F=2。正方体有8个顶点、12条棱、6个面，所以8-12+6=2；三棱柱有6个顶点、9条棱、5个面，所以6-9+5=2。（1）一个四棱锥有5个顶点、8条棱，求它的面数；（2）一个凸多面体有9个顶点、16条棱，求它的面数；（3）用“每一条棱连接两个顶点，每一个面由若干条棱围成”的思想，说明在数多面体的顶点、棱、面时为什么要按统一标准计数。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________26.（9分）综合证明：在△ABC中，AB=AC，点D、E分别在边AB、AC上，且AD=AE，连接BE、CD，BE与CD交于点P。（1）证明△ABE≌△ACD；（2）证明PB=PC；（3）若∠A=40°，且∠ABE=30°，求∠BPC的度数。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析说明：以下答案按题号对应。解答题评分可按关键步骤给分，过程正确且表达清楚即可得相应分值。题号12345678910答案BABBCBCBBB题号11：9412：8013：异面直线14：5015：616：2√31.答案：B解析：把2所在面作为前面，1与3分别折到左右两个相对位置，4折到2的背面，5与6折到上下两个相对位置，因此与1相对的是3。2.答案：A解析：表面最短路程可把相邻的两个面展开。三种可能距离分别为√[(6+4)²+3²]=√109，√[(6+3)²+4²]=√97，√[(4+3)²+6²]=√85，最小为√85cm。3.答案：B解析：三角形内角和为180°，所以∠C=180°-40°-60°=80°。与∠C相邻的外角等于180°-80°=100°，也等于∠A+∠B。4.答案：B解析：已知两边AB、AC分别等于DE、DF，且夹角∠A=∠D，满足两边及其夹角对应相等，所以用SAS判定全等。5.答案：C解析：平行线被截，内错角相等，已知其中一个内错角为72°；与它同旁的内角与72°互补，所以为180°-72°=108°。6.答案：B解析：从正上方看只看底面占据的位置，叠在拐角上方的小正方体不增加新的底面格。底层“L”形共有3个位置，因此看到3个小正方形。7.答案：C解析：在同一平面内，两条直线都平行于第三条直线时，这两条直线互相平行。A项缺少同一平面条件，B项相等角不一定是对顶角，D项两边相等不能判定三角形全等。8.答案：B解析：利用平方差公式(a+b)(a-b)=a²-b²，取a=x，b=2，得(x+2)(x-2)=x²-4。9.答案：B解析：由2x-3<5得2x<8，解得x<4。不等式两边同除以正数2，方向不改变。10.答案：B解析：三角形第三边满足|7-5|<x<7+5，即2<x<12。整数x可以为3、4、5、6、7、8、9、10、11，共9个。11.答案：94解析：长方体表面积S=2(lw+lh+wh)=2(5×4+5×3+4×3)=2(20+15+12)=94cm²。12.答案：80解析：三个内角总份数为2+3+4=9，每份为180°÷9=20°，最大角为4份，即80°。13.答案：异面直线解析：AB在底面ABCD内，CC₁是竖直棱，二者不平行且不相交，不能同处一个平面，所以是异面直线。14.答案：50解析：设∠2=x，则∠1=3x-20°。由互补得x+3x-20°=180°，解得4x=200°，x=50°。15.答案：6解析：A、B两点纵坐标相同，线段AB平行于x轴，长度等于横坐标差的绝对值：|4-(-2)|=6。16.答案：2√3解析：正方体体对角线d满足d²=2²+2²+2²=12，所以d=√12=2√3cm。17.答案：化简结果为-6x+13；当x=1/2时，原式=10。解析：先展开：(x-3)²=x²-6x+9，(x+2)(x-2)=x²-4。两式相减得x²-6x+9-x²+4=-6x+13。把x=1/2代入，得-6×1/2+13=-3+13=10。评分要点：正确展开2分，正确合并2分，代入求值2分。18.答案：AB=AC。解析：因为D是BC的中点，所以BD=DC；又AD⊥BC，所以∠ADB=∠ADC=90°；公共边AD=AD。于是△ADB与△ADC满足两边及夹角对应相等，可由SAS判定△ADB≌△ADC，因此对应边AB=AC。也可用直角三角形中两直角边对应相等判定。19.答案：（1）剩余体积为136cm³。（2）该几何体是高为4cm的直角形柱体，侧面面积可按底面周长乘高计算。解析：原长方体体积为8×5