4.5.1等腰三角形的性质 课件 -2026-2027学年湘教版数学八年级上册

#湘教版数学八年级上册4.5.1等腰三角形的性质专项练习##知识点回顾**等腰三角形定义**：有两条边相等的三角形叫做等腰三角形。相等的两边叫**腰**\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)另一边叫**底边**；两腰的夹角叫**顶角**\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)腰和底边的夹角叫**底角**。###等腰三角形性质1.**性质1（等边对等角）**：等腰三角形的两个底角相等。简写：**等边对等角**。>在$\triangleABC$中\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)若$\(AB=AC\)$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)则$\boldsymbol{\angleB=\angleC}$。2.**性质2（三线合一）**：等腰三角形**顶角平分线\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)底边上的中线\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)底边上的高**互相重合。>只要满足其中一个条件\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)另外两个自动成立。>例：$\(AB=AC\)$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$AD$平分$\angleBAC$$\Rightarrow$$AD\perpBC$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$\(BD=DC\)$。3.对称性：等腰三角形是**轴对称图形**\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)对称轴是**顶角平分线所在的直线（底边上的高/中线所在直线）**。✅解题要点1.已知等腰三角形一个内角\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)要**分类讨论**：这个角是顶角还是底角；底角一定是锐角\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)不能为钝角或直角；2.“三线合一”是证明线段相等\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)角相等\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)垂直的重要工具；3.等边三角形是特殊的等腰三角形。满分：100分时间：40分钟##一\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)选择题（每题4分\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)共20分）1.等腰三角形两个底角相等\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)这个性质叫（）A.等角对等边B.等边对等角C.三线合一D.轴对称2.等腰三角形的对称轴是（）A.底边上的高B.顶角平分线所在直线C.腰上的中线D.底边3.等腰三角形顶角为$80^\circ$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)则底角为（）A.$80^\circ$B.$50^\circ$C.$40^\circ$D.$100^\circ$4.等腰三角形一个内角是$100^\circ$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)则另外两个角是（）A.$100^\circ\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)40^\circ$B.$40^\circ\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)40^\circ$C.$50^\circ\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)50^\circ$D.$100^\circ\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)100^\circ$5.在$\triangleABC$中\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$\(AB=AC\)$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$AD$是底边$BC$上的中线\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)则下列结论不一定成立的是（）A.$AD\perpBC$B.$AD$平分$\angleBAC$C.$\(AB=BC\)$D.$\(BD=CD\)$##二\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)填空题（每题4分\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)共20分）1.等腰三角形的两个底角相等\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)简称________。2.等腰三角形顶角平分线\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)底边上的中线\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)底边上的高________\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)简称三线合一。3.等腰三角形一个底角是$70^\circ$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)顶角是________。4.等腰三角形是________图形\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)对称轴有1条。5.若等腰三角形两边长为$3$和$7$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)则周长为________。##三\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)基础解答题（每题8分\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)共32分）1.已知：在$\triangleABC$中\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$\(AB=AC\)$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$\angleB=65^\circ$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)求$\angleA$的度数。2.已知：$\(AB=AC\)$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$AD\perpBC$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)垂足为$D$。求证：$AD$平分$\angleBAC$。3.已知等腰三角形一个内角为$50^\circ$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)求其余两个内角。4.判断对错\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)并说明理由：等腰三角形的高\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)中线\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)角平分线三线合一。##四\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)拓展提升题（每题14分\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)共28分）1.已知：如图\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$\(AB=AC\)$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)点$D$在$BC$上\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)且$\(BD=CD\)$。求证：$AD\perpBC$。2.已知：$\(AB=AC\)$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)点$D\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)E$在$BC$上\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$\(AD=AE\)$。求证：$\(\(\boldsymbol{BD=CE}\)\)$。---#参考答案与解析##一\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)选择题1.B2.B3.B4.B5.C##二\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)填空题1.**等边对等角**2.**互相重合**3.$\boldsymbol{40^\circ}$4.**轴对称**5.$\boldsymbol{\(\boldsymbol{17}\)}$（解析：$\(3,3,7\)$不能构成三角形\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)只能$\(3,7,7\)$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)周长$3+7+7=\(\boldsymbol{17}\)$）##三\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)基础解答题1.解：$\because\(AB=AC\)$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$\therefore\angleB=\angleC=65^\circ$（等边对等角）$\angleA=180^\circ-\angleB-\angleC=180^\circ-65^\circ-65^\circ=\boldsymbol{50^\circ}$2.证明：$\because\(AB=AC\)\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)AD\perpBC$$\thereforeAD$平分$\angleBAC$（等腰三角形三线合一）3.解：分两种情况①当$50^\circ$是顶角时：底角$=(180^\circ-50^\circ)\div2=65^\circ$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)另外两角：$65^\circ\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)65^\circ$②当$50^\circ$是底角时：顶角$=180^\circ-50^\circ-50^\circ=80^\circ$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)另外两角：$50^\circ\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)80^\circ$综上：另外两角为$65^\circ\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)65^\circ$或$50^\circ\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)80^\circ$4.×；理由：必须强调**底边上**的高\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)中线和**顶角**平分线才三线合一；腰上的高\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)中线不满足这个性质。##四\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)拓展提升题1.证明：$\because\(AB=AC\)\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)\(BD=CD\)$$\thereforeAD\perpBC$（等腰三角形三线合一）2.证明：过点$A$作$AF\perpBC$于$F$。$\because\(AB=AC\)\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)AF\perpBC$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$\thereforeBF=CF$（三线合一）$\because\(AD=AE\)\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)AF\perpBC$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)$\thereforeDF=EF$（三线合一）$\thereforeBF-DF=CF-EF$\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)即$\boldsymbol{\(\(\boldsymbol{BD=CE}\)\)}$##易错警示1.“三线合一”一定是**顶角平分线\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)底边上中线\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)底边上高**\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)不要笼统说成三角形的高\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)中线\(D\(65^\circ\(50^\circ、80^\circ\)65^\circ\)E\)角平分线；2.等腰三角形给一个角求另外两角\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)**一定要分类讨论**；底角不能≥90°；3.等腰三角形边长问题也要分类\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)最后用**三角形三边关系**检验能否组成三角形。下一节：**4.5.2等腰三角形的判定**\(100^\circ\(40^\circ\(50^\circ\(100^\circ\(\becauseAB=AC\(65^\circ\(50^\circ\(\becauseAB=AC\(\becauseAB=AC\(\becauseAD=AE，AF\perpBC\)AF\perpBC\)BD=CD\)80^\circ\)65^\circ\)AD\perpBC\)100^\circ\)50^\circ\)40^\circ\)40^\circ\)继续同模板吗？湘教版（新教材）数学八年级上册4.5.1等腰三角形的性质授课教师：.

班

级：

8年级（

）班.

时

间：.

2026/10/10

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4章三角形学

习

目

标123了解等腰三角形的有关概念.掌握等腰三角形的性质.（重点）能运用等腰三角形的性质进行有关的证明和计算..（难点）导入新课ACB腰腰底边顶角底角底角两条边相等的三角形叫作等腰三角形.

02新知导入回顾2：什么是等腰三角形？两条边相等的三角形叫作等腰三角形.在等腰三角形中，相等的两边叫作腰，另外一边叫作底边.两腰的夹角叫作顶角，腰和底边的夹角叫作底角.02新知导入一般三角形等腰三角形两条边相等等腰三角形作为一种特殊三角形，具有一般三角形的所有性质。03新知探究思考在等腰△ABC中，已知AB=AC，AD是△ABC的中线，则∠B=∠C吗？∠BAD=∠CAD吗？AD是△ABC的高线吗？BD=DC边边边

03新知探究思考在等腰△ABC中，已知AB=AC，AD是△ABC的中线，则∠B=∠C吗？∠BAD=∠CAD吗？AD是△ABC的高线吗？∴∠B=∠C，∠BAD=∠CAD，∠ADB=∠ADC=90°.即AD是△ABC的顶角∠BAC的平分线，是底边BC上的高线.03新知探究AD是△ABC的中线AD是△ABC的顶角∠BAC的平分线AD是底边BC上的高线等腰三角形的性质定理：1.等腰三角形的两个底角相等（简称“等边对等角”).2.底边上的高线、中线及顶角平分线重合（简称“三线合一”).03新知探究找一找：你能找出等腰三角形的对称轴吗？等腰三角形的性质定理：3.等腰三角形是轴对称图形，对称轴是顶角平分线所在的直线.03新知探究如图，在△ABC中，AB=AC，D为AB的中点，点E在AC上，例1解：(1)因为BE=AE，D为AB的中点，所以ED是等腰△EAB的边AB上的中线，从而ED⊥AB(三线合一)且BE=BC=AE.(1)求证：ED⊥AB；(2)求△ABC各角的度数.03新知探究(2)因为AB=AC，BE=BC=AE，所以∠ABC=∠C，∠C=∠1，∠A=∠2(等边对等角).于是∠1=∠A+∠2=2∠A，从而∠ABC=∠C=∠1=2∠A.又∠A+∠ABC+∠C=180°，于是∠A+2∠A+2∠A=180°，从而∠A=36°，因此∠A，∠ABC，∠C的度数分别为36°，72°，72°.

D

返回返回2．如图，B，C两点分别在直线a，b上，且a∥b，BA＝BC，AB⊥a，若∠1＝115°，则∠BCA的度数等于________．25°3.

如图，在△ABC中，AB＝AC，点D，E分别在AC，AB边上，且BC＝BD，AD＝DE＝EB，则∠A的度数为________．返回45°03新知探究议一议如图所示的三角测平架中，AB=AC，在BC的中点D挂一个重锤，自然下垂，调整架身，使点A恰好在铅垂线上.（1）AD与B