14.3.1 角的平分线的性质-课件-2026-2027学年人教版数学八年级上册

#人教版八年级上册14.3.1角的平分线的性质专项练习（全文约1500字\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)可直接打印）##知识要点速记1.**角平分线的性质定理**：角平分线上的点到角两边的距离相等。符号语言：$\becauseOC$平分$\angleAOB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)点$P$在$OC$上\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PD\perpOA\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)PE\perpOB$$\therefore\(PD=PE\)$⚠️关键点：**距离**指垂线段的长\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)必须有垂直！不垂直的线段不适用这个性质。2.回顾：尺规作角平分线\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)原理是SSS全等。3.常用隐含条件-角平分线；-两条垂直（垂线段）；-公共边\(BM、CN\)直角相等。4.解题思路：看到角平分线+点向两边作垂线\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)直接得两条垂线段相等\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)不用再证全等；如果要证明这个性质本身\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)需要用AAS证明两个直角三角形全等。⚠️易错提醒①没有垂直\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)不能直接说线段相等；②性质是：**角平分线上的点$\Rightarrow$距离相等**；不要和后面角平分线判定（距离相等$\Rightarrow$在角平分线上）搞混。##一\(BM、CN\)基础过关（判断+填空+选择）###判断题（对√\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)错×）1.角平分线上任意一点到角两边的线段相等。（）2.角平分线上的点到角两边的垂线段长度相等。（）3.若$PD\perpOA\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)PE\perpOB\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)\(PD=PE\)$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)直接能用角平分线性质得到$OC$平分$\angleAOB$。（）4.一条射线是角平分线\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)上面的点向两边作垂线\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)垂线段相等。（）5.到角两边距离相等的点\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)一定在这个角的平分线上（本课时只学性质\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)这个是判定）。（）###填空题6.角平分线上的点到角两边的________相等。7.已知$OC$平分$\angleAOB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)点$P$在$OC$上\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PM\perpOA$于$M$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PN\perpOB$于$N$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)若$\(PM=4\)$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)则$\(PN=\)$____。8.使用角平分线性质\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)必须同时满足两个条件：①点在________；②向角两边作________。9.角平分线性质定理的证明\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)一般通过证明两个直角三角形________（填判定）全等完成。10.$\angleAOB=60^\circ$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$OC$平分$\angleAOB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)点$P$在$OC$上\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PD\perpOB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)则$\angleDOP=$____$^\circ$。###选择题11.如图\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$OP$平分$\angleAOB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PC\perpOA\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)PD\perpOB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)则一定成立的是（）A.$\(PC=PD\)$B.$\(OC=OP\)$C.$\angleAOP=\angleAOB$D.$\(PC=OP\)$12.下列条件\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)能够直接用角平分线性质得到线段相等的是（）A.点在角平分线上\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)连接角两边上任意两点B.点在角平分线上\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)向两边作垂线段C.两条线段相等\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)则射线平分角D.角里面任意一点到两边距离相等13.在$\triangleABC$中\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$AD$平分$\angleBAC$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$DE\perpAB$于$E$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$DF\perpAC$于$F$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\(DE=5\)$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)则$DF$等于（）A.3B.4C.5D.614.如图\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$OC$平分$\angleAOB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)下列说法错误的是（）A.若$PM\perpOA\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)PN\perpOB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)则$\(PM=PN\)$B.$\angleAOC=\angleBOC$C.只要$\(PM=PN\)$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)就一定有$OC$平分$\angleAOB$D.$OC$是一条射线##二\(BM、CN\)中档解答题15.已知：$OC$平分$\angleAOB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)点$P$在$OC$上\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PD\perpOA$于$D$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PE\perpOB$于$E$。求证：$\(PD=PE\)$。16.已知：在$\triangleABC$中\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$BD$平分$\angleABC$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$DE\perpAB$于$E$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$DF\perpBC$于$F$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\(DE=6\)$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)求$DF$的长。17.已知：$\triangleABC$中\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\angleC=90^\circ$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$AD$平分$\angleCAB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$DE\perpAB$于$E$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\(AC=6\)$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\(CD=4\)$。求$DE$的长。18.已知：$\(AB=AD\)$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$AC$平分$\angleBAD$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$CE\perpAD$于$E$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$CF\perpAB$于$F$。求证：$\(CE=CF\)$。##三\(BM、CN\)拓展提升题19.已知：$\triangleABC$的角平分线$BM\(BM、CN\)CN$交于点$P$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PD\perpBC$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PE\perpAB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PF\perpAC$。求证：$\(PD=PE\)=PF$。20.已知：在$\triangleABC$中\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\angleC=90^\circ$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$AD$平分$\angleBAC$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$DE\perpAB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\(BC=8\)$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\(BD=5\)$。求$DE$的长度。##参考答案1.×（必须是垂线段）2.√3.×（这是角平分线的判定\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)不是性质）4.√5.√（这句话本身是判定\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)本节课不使用）6.距离（垂线段长）7.$4$8.角平分线上；垂线（垂线段）9.AAS10.$30$11.A12.B13.C14.C15.证明：$\becauseOC$平分$\angleAOB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\therefore\angleDOP=\angleEOP$$\becausePD\perpOA\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)PE\perpOB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\therefore\angleODP=\angleOEP=90^\circ$在$\triangleODP$和$\triangleOEP$中$\begin{cases}\angleODP=\angleOEP\\\angleDOP=\angleEOP\\OP=OP\end{cases}$$\therefore\triangleODP\cong\triangleOEP(\text{AAS})$$\therefore\(PD=PE\)$16.解：$\becauseBD$平分$\angleABC$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$DE\perpAB\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)DF\perpBC$$\thereforeDF=\(DE=6\)$答：$\(DF=6\)$。17.解：$\becauseAD$平分$\angleCAB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\angleC=90^\circ$即$DC\perpAC$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$DE\perpAB$$\thereforeDE=\(CD=4\)$答：$\(DE=4\)$。18.证明：$\becauseAC$平分$\angleBAD$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$CE\perpAD\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)CF\perpAB$根据角平分线的性质$\therefore\(CE=CF\)$19.证明：$\becauseBM$平分$\angleABC$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PD\perpBC\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)PE\perpAB$$\thereforePE=PD$$\becauseCN$平分$\angleACB$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$PD\perpBC\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)PF\perpAC$$\thereforePD=PF$$\therefore\(PD=PE\)=PF$20.解：$\because\(BC=8\)\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)\(BD=5\)$$\thereforeCD=BC-BD=8-5=3$$\becauseAD$平分$\angleBAC$\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)$\angleC=90^\circ\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)DE\perpAB$$\thereforeDE=CD=3$答：$\(DE=3\)$。##解题小结1.性质口诀：**角平分线上的点\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)到角两边垂线段相等**；没有垂直不能用；2.看到角平分线+双垂直\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)优先直接用性质\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpBC\(PD\perpBC\(\becauseBC=8\(\angleC=90^\circ，DE\perpAB\)BD=5\)PF\perpAC\)PE\perpAB\)CF\perpAB\)DF\perpBC\)PE\perpOB\)PN\perpOB\)PD\perpOB\)PE\perpOB\)省去全等证明\(PD\perpOA\(PC\perpOA\(PM\perpOA\(\becausePD\perpOA\(DE\perpAB\(CE\perpAD\(PD\perpB