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新课导入生活中的相交直线生活中的相交直线有一个公共点的两条直线形成相交直线．相交线的定义●O1234知识要点二线四角图请你画出任意两条相交直线，看看这四个角有什么关系?两条相交直线形成的小于平角的角有几个?如图1所示，∠1与∠2有什么特点？∠1与∠2有一条公共边OA，它们的另一边互为反向延长线．2314ABCDOshop.99114.com/50299198shop.99114.com/50646179shop.99114.com/50627572shop.99114.com/50674705shop.99114.com/50798018shop.99114.com/50662906shop.99114.com/50499147shop.99114.com/50657920shop.99114.com/50672964shop.99114.com/51126331shop.99114.com/51232986shop.99114.com/51129260shop.99114.com/51233446shop.99114.com/51131343shop.99114.com/51132188shop.99114.com/51126246shop.99114.com/49830982shop.99114.com/50855587shop.99114.com/51132012shop.99114.com/50857031shop.99114.com/50855990shop.99114.com/51023652shop.99114.com/51233435shop.99114.com/50866272shop.99114.com/51233066shop.99114.com/51004630shop.99114.com/51028921shop.99114.com/50798552shop.99114.com/51233029shop.99114.com/51235754shop.99114.com/50673234shop.99114.com/50673283shop.99114.com/50672593shop.99114.com/50847408shop.99114.com/50674016shop.99114.com/50857820shop.99114.com/50823805shop.99114.com/50802033shop.99114.com/51130583shop.99114.com/49610836shop.99114.com/51131153shop.99114.com/50668455shop.99114.com/50831388shop.99114.com/51239193shop.99114.com/50673281shop.99114.com/50673437shop.99114.com/50673209shop.99114.com/50669256shop.99114.com/51399270shop.99114.com/51396689shop.99114.com/51398726shop.99114.com/51233072shop.99114.com/50657958shop.99114.com/51405322shop.99114.com/41244405shop.99114.com/51407253shop.99114.com/51023949shop.99114.com/51408617shop.99114.com/51410378shop.99114.com/51410288shop.99114.com/51238418shop.99114.com/51407265shop.99114.com/48001267shop.99114.com/50637206shop.99114.com/51410820shop.99114.com/50498383shop.99114.com/51413017shop.99114.com/49312439shop.99114.com/51413260shop.99114.com/48900764shop.99114.com/51397406shop.99114.com/51410598shop.99114.com/51410528shop.99114.com/48211017shop.99114.com/41321718shop.99114.com/48795003shop.99114.com/51413553shop.99114.com/51399142shop.99114.com/51414416shop.99114.com/50673484shop.99114.com/50673486shop.99114.com/50673407shop.99114.com/51413290shop.99114.com/51413954shop.99114.com/47915143shop.99114.com/50673278shop.99114.com/51413940shop.99114.com/51413944shop.99114.com/51416024shop.99114.com/51415580shop.99114.com/51416036shop.99114.com/50658016shop.99114.com/50648437shop.99114.com/51417568shop.99114.com/50659687shop.99114.com/51418705shop.99114.com/50659865shop.99114.com/51413948shop.99114.com/51417271shop.99114.com/49826699如果两个角有一条公共边，它们的另一边互为反向延长线，那么这两个角互为邻补角．图中互为邻补角的有：∠1与∠2，∠2与∠3，∠3与∠4，∠1与∠4．知识要点邻补角OCDAB1342判断两个角是不是邻补角：（1）有一个公共顶点；（2）有一条公共边．一对邻补角一定互补吗？一对互补的角一定是邻补角吗？图中，∠1和∠2、∠2和∠3、∠3和∠4，∠1和∠4都是邻补角，它们是相互的、成对出现的，如∠2是∠3的邻补角，∠1是∠4的邻补角，单独的一个∠1或单独的一个∠4都不能叫邻补角．OCDAB1342CA1OOCB4下列图中∠1、∠2还是邻补角吗？121212如图1所示，∠1与∠3有什么特点？∠1与∠3是直线AB与CD相交得到的，它们有一个公共顶点O，没有公共边．2314ABCDO如果一个角的两边是另一个角的两边的反向延长线，那么这两个角互为对顶角．右图中互为对顶角的为：∠1与∠3；∠2与∠4．知识要点对顶角OCDAB1342判断两个角是不是对顶角：（1）两个角是由两条直线相交而形成的（由两条直线相交保证了所形成的角有公共顶点）；（2）两个角的两边无公共边．下列各图中∠1、∠2是对顶角吗？为什么？211212对顶角是成对出现的．上图中，∠2和∠4它们是相互的，∠2是∠4的对顶角，∠4是∠2的对顶角，而单独的一个∠2或一个单独的∠4都不能叫对顶角．OCDAB1342OA2DOCB4两条直线相交，有____组对顶角．三条直线相交于一点，有____组对顶角．26四条直线相交于一点，有____组对顶角．ｎ条直线相交于一点，有_________组对顶角．12n（n－1）∠1与∠2互补，∠2与∠3互补∠1＝∠3（同角的补角相等）∠2＝∠4对顶角相等．OCDAB1342知识要点对顶角性质对顶角相等.例：如图所示，直线m，n相交，∠1＝60°，求∠2，∠3，∠4的度数．mn1234解：由邻补角的定义，可得：∠2＝180°－∠1＝180°－60°＝120°；由对顶相等，可得：∠3＝∠1＝60°，∠4＝∠2＝120°．角的名称特征性质相同点不同点对顶角邻补角对顶角相等邻补角互补②有公共顶点;③没有公共边①两条直线相交形成的角；①两条直线相交而成；②有公共顶点;③有一条公共边①都是两条直线相交而成的角；③都是成对出现的②都有一个公共顶点；②两直线相交时，对顶角只有两对邻补角有四对①有无公共边课堂小结（1）有公共顶点且相等的两个角是对顶角．（）（2）两条直线相交，有两组对顶角．（）1．判断×√随堂练习2．如右图直线AB、CD交于点O，OP为射线，那么（）A．∠AOC和∠BOC是对顶角B．∠BOC和∠AOP是对顶角C．∠BOC和∠AOD是对顶角D．∠AOC和∠DOP是对顶角CDABOPC3．如图，直线a，b相交于点O，若∠1＝40°，则∠2＝（）A．60°B．100°C．120°D．140°12OabDCDABOP2．直线AB、CD交于点O，OP是∠BOC的平分线，已知∠AOC=54°．求∠BOP的度数．解：由邻补角的定义可得：∠BOC＝180°－∠AOC＝180°－54°＝126°；因为OP平分∠BOC，所以∠BOP=∠AOD=×126°＝63°.1212生活中的垂线生活中的垂线当∠BOD＝90°时．∠AOD＝_______；∠AOC＝_______；∠BOC＝_______；BAODC此时我们说，AB与CD互相垂直．90°90°90°当∠BOD＝α°（α≠90°）时．∠AOD＝（180－α）°∠AOC＝（α）°∠BOC＝（180－α）°BAODC当α≠90°时，AB与CD不垂直，此时我们说AB与CD斜交．两条直线相交斜交垂直——相交的特殊情况┓当两条直线相交所成的四个角中，有一个角是直角时，这两条直线互相垂直，其中一条直线叫另一条直线的垂线，它们的交点叫垂足．垂直图中m与n互相垂直，其中，m叫n的垂线，n叫m的垂线，垂足为O．知识要点nmO┓用“⊥”和直线字母表示垂直垂直的表示：例如，如图，m、n互相垂直，垂足为O，则记为：m⊥n或n⊥m．若要强调垂足，则记为：a⊥b，垂足为O．nmO┓书写形式1：如图，当直线AB与CD相交于O点，∠AOD=90°时，AB⊥CD，垂足为O．因为∠AOD=90°（已知）所以AB⊥CD（垂直的定义）书写形式2：反之，若直线AB与CD垂直，垂足为O，那么，∠AOD=90°．垂直的书写形式：因为AB⊥CD（已知）所以∠AOD=90°（垂直的定义）应用垂直的定义：∠AOC=∠BOC=∠BOD=90°ABCDO判断两条直线互相垂直的关键：只要找到两条直线相交时四个交角中一个角是直角．生活中常见的互相垂直的例子例2如图．直线AB、CD相交于点O，OE⊥AB于O，OB平分∠DOF，∠DOE=50°，求∠AOC、∠EOF、∠COF的度数．解:因为AB⊥OE（已知）所以∠EOB=90°(垂直的定义)因为∠DOE=50°（已知）所以∠DOB=40°(互余的定义)所以∠AOC=∠DOB=40°（对顶角相等）又因为OB平分∠DOF所以∠BOF=∠DOB=40°（角平分线定义）所以∠EOF=∠EOB+∠BOF=90°+40°=130°所以∠COF=∠COD－∠DOF=180°－80°=100°(邻补角定义)ACEBDOF如图，直线AB、CD相交于点O，OE⊥AB，∠1=125°，求∠COE的度数．练一练ACEBDO135°垂线的定义定义图示文字语言几何语言两层含义当两条直线所成的四个角中有一个角是直角时,我们就说这两条直线互相垂直．直线AB垂直于直线CD，O为垂足．AB⊥CD，O为垂足．含义1：∵AB⊥CD∴∠1=90°含义2：∵∠1=90°∴AB⊥CD（垂直用符号“⊥”来表示，读作“垂直于”）1OABCD┓如图,CD⊥EF,∠1=∠2,则AB⊥EF．请说明理由（补全解答过程）90°⊥垂线的定义垂线的定义90°练一练解：∵CD⊥EF（已知）∴∠1=____()∵∠1=∠2=____∴AB___EF()EABCDF12EC1例：如图，直线AB与直线CD相交于点O，OE⊥AB，已知∠BOD=45°，求∠COE的度数．解：因为OE