22.3.2相似三角形的判定定理1 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22.3.2相似三角形的判定定理1（两角分别相等的两个三角形相似）##一\(AB\(AC\(AB、CD\)BD\)CD\)知识点梳理###判定定理1：**两角分别相等的两个三角形相似**简单记：**AA（两角对应相等）→三角形相似**>内容：如果一个三角形的两个角分别与另一个三角形的两个角对应相等\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)那么这两个三角形相似。原理：三角形内角和180°\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)只要2组角相等\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)第三组角自然相等。书写：若$\angleA=\angleD,\angleB=\angleE$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)则$\boldsymbol{\triangleABC\backsim\triangleDEF}$✅常用模型（必考）1.**A字型**：$DE\parallelBC$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleADE=\angleABC\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)\angleAED=\angleACB$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)AA证相似（就是前面的预备定理）2.**8字型（对顶角模型）**：对顶角相等\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)再加一组角相等\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)AA相似3.**母子型（直角三角形射影基本图）**：Rt△斜边上的高\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)分出两个小直角三角形都和原大三角形两两相似（三组相似\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)都是AA）⚠易错提醒1.必须是**对应角相等**\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)不是随便两个角；2.只要找到**两组角对应相等**就足够\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)不用再找边；3.直角三角形：只要再有一组锐角相等\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)就可以用AA判定相似；4.书写相似时\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)**对应顶点对齐**。##二\(AB\(AC\(AB、CD\)BD\)CD\)选择题1.在$\triangleABC$和$\triangleDEF$中\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleA=\angleD=50^\circ$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleB=\angleE=60^\circ$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)则（）A.$\triangleABC\backsim\triangleDEF$B.$\triangleABC\backsim\triangleDFE$C.$\triangleABC\backsim\triangleEFD$D.无法判定2.下列各组三角形\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)一定相似的是（）A.两个等腰三角形\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)都有一个角40°B.两个直角三角形\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)都有一个锐角30°C.两个等腰直角三角形\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)斜边相等D.两个钝角三角形\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)都有一个钝角100°3.如图\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$AB\(AB\(AC\(AB、CD\)BD\)CD\)CD$交于点$O$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleA=\angleC$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)则相似的是（）A.$\triangleAOD\backsim\triangleCOB$B.$\triangleAOD\backsim\triangleBOC$C.$\triangleAOB\backsim\triangleCOD$D.$\triangleAOB\backsim\triangleDOC$4.在$\triangleABC$中\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleACB=90^\circ$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$CD\perpAB$于$D$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)下面不能用AA证明相似的是（）A.$\triangleACD\backsim\triangleABC$B.$\triangleBCD\backsim\triangleBAC$C.$\triangleACD\backsim\triangleBCD$D.$\triangleACD\backsim\triangleCBD$5.已知$\triangleABC$与$\triangleMNP$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleA=\angleM$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)再添加一个条件\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)能判定相似的是（）A.$\(AB=MN\)$B.$\angleB=\angleN$C.$\(BC=NP\)$D.$\(AC=MP\)$##三\(AB\(AC\(AB、CD\)BD\)CD\)填空题1.两个三角形\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)有两组角对应相等\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)则这两个三角形________。2.在$\triangleABC$和$\triangleDEF$中\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleA=\angleD$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleC=\angleF$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)则$\triangleABC\backsim$________。3.直角三角形中\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)只要有一组________相等\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)两个直角三角形就相似（AA）。4.$\triangleABC$中\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleA=70^\circ,\angleB=50^\circ$；$\triangleDEF$中$\angleD=70^\circ,\angleF=60^\circ$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)则两三角形______（填相似/不相似）。5.8字型模型：$AC\(AB\(AC\(AB、CD\)BD\)CD\)BD$交于$O$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleA=\angleB$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleAOC=\angleBOD$（对顶角）\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)所以$\triangleAOC\backsim$________。##四\(AB\(AC\(AB、CD\)BD\)CD\)解答题1.已知：在$\triangleABC$和$\triangleA'B'C'$中\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleA=\angleA'=60^\circ$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleB=\angleB'=40^\circ$。求证：$\triangleABC\backsim\triangleA'B'C'$。2.如图\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$DE\parallelBC$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)求证：$\triangleADE\backsim\triangleABC$（用AA判定）。3.已知：$AB\(AB\(AC\(AB、CD\)BD\)CD\)CD$相交于$O$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleA=\angleC$。求证：$\triangleAOD\backsim\triangleCOB$。4.拓展：在Rt\(\triangleABC\)\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleACB=90^\circ$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$CD\perpAB$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)垂足$D$。求证：$\triangleACD\backsim\triangleABC$。##五\(AB\(AC\(AB、CD\)BD\)CD\)参考答案与解析###选择题1.A解析：两组角对应相等\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)AA相似\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)顶点一一对应。2.B解析：都有直角+30°锐角\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)两组角相等\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)一定相似。3.D解析：$\angleA=\angleC\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)\angleAOB=\angleCOD$（对顶角）\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)AA\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\triangleAOB\backsim\triangleCOD$。4.C解析：C选项顶点不对应\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)D是正确写法$\triangleACD\backsim\triangleCBD$。5.B解析：增加一组角相等\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)满足AA判定。###填空题1.相似2.$\triangleDEF$3.锐角4.相似（$\angleC=60^\circ$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleF=60^\circ$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)两组角相等）5.$\triangleBOD$###解答题1.证明：$\because\angleA=\angleA'=60^\circ\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)\angleB=\angleB'=40^\circ$$\therefore\triangleABC\backsim\triangleA'B'C'$（两角分别相等的两个三角形相似）2.证明：$\becauseDE\parallelBC$$\therefore\angleADE=\angleABC$（同位角相等）又$\because\angleA=\angleA$（公共角）$\therefore\triangleADE\backsim\triangleABC$（AA）3.证明：$\because\angleA=\angleC$\(\angleADE=\angleABC\(\angleA=\angleC\(\because\angleA=\angleA'=60^\circ\(\because\angleACB=90^\circ，CD\perpAB\)\angleB=\angleB'=40^\circ\)\angleAOB=\angleCOD\)\angleAED=\angleACB\)$\angleAOD=\angleCOB$（对顶角相等）$\therefore\triangleAOD\backsim\triangleCOB$（AA）4.证明：$\because\angleACB=90^\circ\(\angleADE=\angleABC\(\angleA=\angleC\(\because\ang