22.3.1相似三角形 课件 -2026-2027学年华东师大版数学九年级上册

#华东师大版九年级上册22.3.\(\boldsymbol{1}\)《相似三角形》专项练习##一\(AB\(D、E\)AC\)知识点梳理###\(\boldsymbol{1}\).相似三角形定义**三个角对应相等\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)三条边对应成比例的两个三角形\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)叫做相似三角形。**-符号：$\boldsymbol{\backsim}$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)读作“相似于”例：$\triangleABC\backsim\triangleA'B'C'$>书写要求：**对应顶点写在对应位置**\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)方便找对应角\(AB\(D、E\)AC\)对应边。###2.相似三角形性质（定义直接推出）\(\boldsymbol{1}\).对应角相等；2.对应边成比例。###3.相似比相似三角形**对应边的比**叫做相似比。>注意：相似比有顺序！$\triangleABC\backsim\triangleA'B'C'$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)相似比$k=\dfrac{AB}{A'B'}$；反过来$\triangleA'B'C'\backsim\triangleABC$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)相似比为$\dfrac\(\boldsymbol{1}\)k$。>全等三角形是相似比$\boldsymbol{\(\boldsymbol{k=\(\boldsymbol{1}\)}\)}$的特殊相似三角形。###4.预备定理（非常重要！平行线证相似）**平行于三角形一边的直线\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)和其他两边（或两边延长线）相交\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)所构成的三角形与原三角形相似。**在$\triangleABC$中\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$DE\parallelBC$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)则$\boldsymbol{\triangleADE\backsim\triangleABC}$⚠本节易错点\(\boldsymbol{1}\).写相似符号时\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)顶点顺序不能乱\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)否则对应边\(AB\(D、E\)AC\)角会找错；2.相似比顺序不能颠倒；3.预备定理前提：**平行线**\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)才能直接得到三角形相似；4.相似三角形只要求角相等\(AB\(D、E\)AC\)边成比例\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)大小可以不一样。##二\(AB\(D、E\)AC\)选择题\(\boldsymbol{1}\).$\triangleABC\backsim\triangleDEF$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$\angleA=50^\circ$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)则$\angleD=$（）A.$40^\circ$B.$50^\circ$C.$60^\circ$D.$\(\boldsymbol{1}\)30^\circ$2.$\triangleABC\backsim\triangleA'B'C'$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$AB=2\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)A'B'=4$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)则$\triangleABC$与$\triangleA'B'C'$的相似比是（）A.$2$B.$\dfrac\(\boldsymbol{1}\)2$C.$\dfrac\(\boldsymbol{1}\)4$D.$4$3.如图\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)在$\triangleABC$中\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$DE\parallelBC$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)交$AB\(AB\(D、E\)AC\)AC$于$D\(AB\(D、E\)AC\)E$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)下列结论正确的是（）A.$\triangleADE\backsim\triangleABC$B.$\triangleADE\backsim\triangleBDE$C.$\triangleADE\backsim\triangleBCD$D.$\triangleADE\backsim\triangleCDE$4.若$\triangleABC\backsim\trianglePQR$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)对应边$\(AB=6\)\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)PQ=3$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)则下面说法错误的是（）A.相似比为$2$（$\triangleABC$到$\trianglePQR$）B.$\angleA=\angleP$C.$\dfrac{BC}{QR}=\dfrac{AB}{PQ}$D.$\dfrac{AC}{QR}=\dfrac{AB}{PQ}$5.下列说法正确的是（）A.两个直角三角形一定相似B.两个等腰三角形一定相似C.两个等边三角形一定相似D.两个等腰直角三角形不一定相似##三\(AB\(D、E\)AC\)填空题\(\boldsymbol{1}\).$\triangleABC\backsim\triangleMNP$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)对应角：$\angleB=$______。2.若$\triangleABC\backsim\triangleA'B'C'$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)相似比为$\(\(\boldsymbol{1}\):3\)$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)则$\triangleA'B'C'$与$\triangleABC$的相似比为______。3.$DE\parallelBC$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$\triangleADE\backsim\triangleABC$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$\(AD:AB=2:5\)$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)则相似比为______。4.$\triangleABC\backsim\triangleDEF$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$AB=4\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)DE=6$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$\(BC=6\)$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)则$\(EF=\)$______。5.全等三角形的相似比等于________。##四\(AB\(D、E\)AC\)解答题\(\boldsymbol{1}\).已知$\triangleABC\backsim\triangleA'B'C'$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$AB=3\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)A'B'=6\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)BC=4$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)求$B'C'$的长。2.在$\triangleABC$中\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$DE\parallelBC$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$AD=2\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)DB=3\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)DE=4$。（\(\boldsymbol{1}\)）求证：$\triangleADE\backsim\triangleABC$；（2）求$BC$的长度。3.$\triangleABC\backsim\triangleDEF$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$\angleA=45^\circ\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)\angleB=60^\circ$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)求$\triangleDEF$中$\angleF$的度数。4.拓展题：$\triangleABC\backsim\triangleA'B'C'$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)相似比$\(2:3\)$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)若$A'B'=9$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)求$AB$的长。##五\(AB\(D、E\)AC\)参考答案与解析###选择题\(\boldsymbol{1}\).B解析：相似三角形对应角相等\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$\angleD=\angleA=50^\circ$2.B解析：$\triangleABC:\triangleA'B'C'=AB:A'B'=2:4=\dfrac\(\boldsymbol{1}\)2$3.A解析：平行线预备定理\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$DE\parallelBC\Rightarrow\triangleADE\backsim\triangleABC$4.D解析：$AC$对应$PR$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)不是$QR$5.C解析：所有等边三角形内角都是60°\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)一定相似。###填空题\(\boldsymbol{1}\).$\angleN$2.$\(3:\(\boldsymbol{1}\)\)$（或$3$）3.$\boldsymbol{\dfrac{2}{5}}$4.$9$解析：$\dfrac{AB}{DE}=\dfrac{BC}{EF},\dfrac46=\dfrac{6}{EF}\Rightarrow\(EF=\)9$5.$\boldsymbol{\(\boldsymbol{1}\)}$###解答题\(\boldsymbol{1}\).解：$\because\triangleABC\backsim\triangleA'B'C'$$\therefore\dfrac{AB}{A'B'}=\dfrac{BC}{B'C'}$$\dfrac{3}{6}=\dfrac{4}{B'C'}$解得$B'C'=8$答：$B'C'=8$。2.解：(\(\boldsymbol{1}\))证明：$\becauseDE\parallelBC$$\therefore\triangleADE\backsim\triangleABC$（平行线预备定理）(2)$AD=2\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)DB=3\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)AB=AD+DB=5$相似比$\dfrac{AD}{AB}=\dfrac{2}{5}$$\dfrac{DE}{BC}=\dfrac25,\dfrac{4}{BC}=\dfrac25$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)解得$BC=\(\boldsymbol{1}\)0$答：$BC=\(\boldsymbol{1}\)0$。3.解：$\triangleABC$内角和$\(\boldsymbol{1}\)80^\circ$$\angleC=\(\boldsymbol{1}\)80^\circ-\angleA-\angleB=\(\boldsymbol{1}\)80^\circ-45^\circ-60^\circ=75^\circ$$\because\triangleABC\backsim\triangleDEF$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)$\therefore\angleF=\angleC=75^\circ$答：$\angleF=75^\circ$。4.解：$\because\triangleABC\backsim\triangleA'B'C'$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)相似比$AB:A'B'=\(2:3\)$$\dfrac{AB}{9}=\dfrac{2}{3}$\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)解得$\(AB=6\)$答：$\(AB=6\)$。---下一节是**22.3.2相似三角形的判定**\(AB=2\(\(AB=6\)\(AB=4\(\angleA=45^\circ，\angleB=60^\circ\)DE=6\)PQ=3\)A'B'=4\)继续吗？华东师大版（新教材）数学九年级上册22.3.1相似三角形授课教师：.

班

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9年级（

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2026/10/11

学子如同向日葵向往暖阳，满心热爱渴求知识。在书页间汲取力量，于思索中追逐光亮，不惧学习路上的磨砺，心怀憧憬奋力向上，向着理想不断成长。第22章图形的相似复习回顾相似多边形的性质和定义？相似多边形的对应边成比例，对应角相等。