作二面角的平面角的常用方法课件

二面角二面角二面角二面角

从空间一直线出发的两个半一、二面角的定义二、二面角的平面角角的平面角

一个平面垂直于二面角的棱，并与两半平面分别相交于射线PA、PB垂足为P，则∠APB叫做二面ABPγβαιαβι平面所组成的图形叫做二面角1、定义二面角二面角从空间一直线出发的两个半一、二面角的定义二、二面角的2、作二面角的平面角的常用方法①、点P在棱上②、点P在一个半平面上③、点P在二面角内ιpαβABABpαβιABOαβιp—定义法—三垂线定理法—垂面法二面角2、作二面角的平面角的常用方法①、点P在棱上②、点P在一个半1、如图，AB是圆的直径，PA垂直圆所在的平面，C是圆上任一点，则二面角P-BC-A的平面角为:A.∠ABPB.∠ACPC.都不是

练习2、已知P为二面角内一点，且P到两个半平面的距离都等于P到棱的距离的一半，则这个二面角的度数是多少？pαβιABOABCP60º二面角1、如图，AB是圆的直径，PA垂直圆所在的平面，C是圆上任一例1.如图，已知P是二面角α-AB-β棱上一点，过P分别在α、β内引射线PM、PN，且∠MPN=60º

∠BPM=∠BPN=45º

，求此二面角的度数。βαABPMNCDO解：在PB上取不同于P的一点O，在α内过O作OC⊥AB交PM于C，在β内作OD⊥AB交PN于D，连CD，可得∠COD是二面角α-AB-β的平面角设PO=a

，∵∠BPM=∠BPN=45º∴CO=a，

DO=a，

PCa，

PDa又∵∠MPN=60º

∴CD=PCa∴∠COD=90º因此，二面角的度数为90ºaOPC二面角例1.如图，已知P是二面角α-AB-β棱上一点，过P分别在α例2．如图P为二面角α–ι–β内一点，PA⊥α,PB⊥β,且PA=5，PB=8，AB=7，求这二面角的度数。

过PA、PB的平面PAB与棱ι

交于O点∵PA⊥α∴PA⊥ι

∵PB⊥β∴PB⊥ι

∴ι⊥平面PAB∴∠AOB为二面角α–ι–β的平面角又∵PA=5，PB=8，AB=7由余弦定理得∴∠P=60º∴∠AOB=120º

∴这二面角的度数为120º解：βαABPιO二面角例2．如图P为二面角α–ι–β内一点，PA⊥α,PB⊥β,且OABPC取AB的中点为E，连PE，OE∵O为AC中点，∠ABC=90º∴OE∥BC且

OEBC在Rt△POE中，OE

，PO∴∴所求的二面角P-AB-C的正切值为例3．如图，三棱锥P-ABC的顶点P在底面ABC上的射影是底面Rt△ABC斜边AC的中点O，若PB=AB=1，BC=，求二面角P-AB-C的正切值。∴∠PEO为二面角P-AB-C的平面角在Rt△PBE中，BE，PB=1，PEOE⊥AB，因此PE⊥ABE解：EOP二面角OABPC取AB的中点为E，连PE，OE∵O为AC中点练习1：已知Rt△ABC在平面α内，斜边AB在30º的二面角α-AB-β的棱上，若AC=5，BC=12，求点C到平面β的距离CO。βαACBOD练习2：在平面四边形ABCD中，AB=BC=2，AD=CD=,∠B=120º；将三角形ABC沿四边形ABCD的对角线AC折起来，使DB′=，求△AB′C所在平面与△ADC所在平面所成二面角的平面角的度数。ABCB’DO二面角练习1：已知Rt△ABC在平面α内，斜边AB在30º的二面角二、二面角的平面角一、二面角的定义

从空间一直线出发的两个半平面所组成的图形叫做二面角1、定义2、求二面角的平面角方法①点P在棱上②点P在一个半平面上③点P在二面角内ABPγβαι小结二面角二面角αβιABαβιpιpαβABpαβιABO—定义法—三垂线定理法—垂面法二面角二、二面角的平面角一、二面角的定义从空间一直线出发的αβABCD

A为二面角α–CD–β的棱CD上一点，AB在平面α内且与棱CD成45º角，又AB与平面β成30º，求二面角α–CD–β的大小。作业二面角二面角二面角αβABCDA为二面角α–CD–β的棱CD上一二面角谢谢！二面角谢谢！初一英语第二学期同步串讲第七讲

Doyouhaveaneraser?初一英语第二学期同步串讲Doyouhaveasoccerball?作二面角的平面角的常用方法课件soccerbasketballbadmintonbaseballballgamessoccerbasketballbadmintonbasetennis(racket)volleyballgolfPingpong(ball)tabletennistennis(racket)volleyballgolfPDoyouhaveanAmericanfootball/rugby?Yes,Ido./No,Idon’t.DoyouhaveanAmericanfootbaGolfclubDoyouhaveagolfball?Yes,Ido./No,Idon’t.GolfclubDoyouhaveagolfbaGroupworkNamesDoyouhavea…?BallsyouhaveLucyYes,Ido.basketballWhatballsdoyouhave?Ihave….GroupworkNamesDoyouhavea…1.你有一个排球吗？是的，我有。2.你有表兄妹吗?不,我没有。3.她有网球拍吗？是的，她有。7.咱们去踢球吧。Doyouhaveavolleyball?Yes,Ido.Doyouhaveanycousins?No,Idon’t.Doesshehaveatennisracket?Yes,shedoes.Let’splaysoccer.1.你有一个排球吗？是的，我有。Doyou4.他有英英字典吗？不，他没有。他有一本英汉字典。5.他们有笔记本吗？是的，他们有。6.让我看一看。DoeshehaveanEnglishEnglishdictionary?No,hedoesn’t.HehasanEnglishChinesedictionary?Dotheyhaveanynotebooks?Yes,theydo.Let’shavealook.4.他有英英字典吗？不，他没有。他有一本英汉字典。DoeDoeshehaveatennisracket?Yes,hedoes./No,hedoesn't.Doeshehaveatennisracket?YDoesshehaveasoccerball?Yes,shedoes./No,shedoesn't.Doesshehaveasoccerball?YeDoeshehaveabasketball?Yes,hedoes.Doeshehaveabasketball?Yes,Dotheyhaveapuck?Yes,theydo./No,theydon’t.icehockeyDotheyhaveapuck?icehockeyA:Iwanttoplaybaseball.Doyouhaveabaseballbat?B:No,Idon’t.A:Doesyourbrotherhaveone?B:Yes,hedoes.Let’sgoandfindhim.A:Iwanttoplaybaseball.DoLet’splayvolleyball.Thatsoundsgood.Let’splayvolleyball.Let’splaybarbell.BarbellThatsoundsboring.Let’splaybarbell.BarbellThatLet’splaybowling.Thatsoundsinteresting.Let’splaybowling.ThatsoundsLet’splaybaseball.Thatsoundsdifficult.Let’splaybaseball.Let’sgoskateboarding.Thatsoundsfun.(exciting)Let’sgoskateboarding.Let’splayPingPong.Thatsoundsrelaxing.letsb.dosth.Let’shaveabreak.Let’sgoandplayPing-pong.Let’splayPingPong.letsb.dOurSchoolSportsCenterneedsnewmembers.Fillintheenrollmentformifyouwanttojointheclubs.Firstname__________Lastname__________Age________________Sex________________Class______________Club_______________Reason__________________________________What’syourfirstname,please?Howoldareyou?Whatclassareyouin?Whatclubdoyouwanttojoin?Why?(Because…Ilike…)OurSchoolSportsCenterneed写出反义词dislike______interesting______good_______well________easy________lost_________day________different_______big________first_________on________infrontof_________this_______these_____________bring______写出反义词Fillintheblanks1.A:Excuseme,_____you____abaseball?B:No,I_____.A:______Jim_____one?B:Yes,he______.Let’sgoandfindhim.2.A:Hi,Sally.B:Hi,Jane.A:Sally,________myfriend,Lucy.B:Hi,Lucy._____tomeetyou.C:Niceto_____you,_____.dohavedon’tDoeshavedoesthisisNicemeettooFillintheblanksdoA:Let’splayvolleyball.B:That______interesting,butI_____haveone.A:Well,_____youhaveatennisbat?B:Yes,I_____.Let’sgoandplaytennis.A:That’sagood______.soundsdon’tdodoideaA:Let’splayvolleyball.soundDoesBhaveaneraser?Istheshopopen?Whatdayistoday?Whatcantheydo?No,shedoesn’t.No,itisn’t.Itisclosed.TodayisFriday.Theygotothesupermarket.Part1DoesBhaveaneraser?No,sheWhatwouldBlike?HowmuchmoneydoesBhave?Whereisherpurse?Whatdoesshedo?

Sheonlyhastwoyuan.Herpurseisathome.SheborrowsfouryuanfromC.Part2She’dlikearedpencilandapencil-sharpener.WhatwouldBlike?Part2She’dDoesJimhaveadictionary?DoesAnnhaveadictionary?WhohasanEnglish-Chinesedictionary?4.WhencanLiLeigiveitback?

Part3No,hedoesn’t.Lucyhasone.HecangiveitbacknextMonday.No,shedoesn’t.DoesJimhaveadictionary?P

close***closed

Theshopclosesat8pm.Pleaseclosethedoor.Thedoorisclosed.Theshopisclosed.Theshopisopen.close***closedThesh

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Thereissomethingwrongwith…close***closedborrYouaregoingtohaveaP.Eclass,thinkaboutsomeactivities.Talkaboutthem.Time_________Place________Sport________Reason________YouaregoingtohaveaP.EclWritingWriteapassageaboutyourselfandyourfriend.Whatsportsdoyouplay?Whatsportsthingsdoyouhave?Whatotherthingsdoyoudo?Whataboutyourfriend?(parent)WritingWriteapassageaboutyKeypoints:

FoodDoyoulikebananas?Doeshe/shelikebananas?Doyoulikebananas?Keypoints:FoodDoyoulikebDoyoulikeoranges?Yes,Ido.No,Idon’t.Doyoulikeoranges?Doeshelikeapples?Yes,hedoes.No,hedoesn't.Doeshelikeapples?Doesshelikepeaches?Yes,shedoes.No,shedoesn't.Doesshelikepeaches?Dotheylikegrapes?Yes,theydo.No,theydon’t.Dotheylikegrapes?Ilikehamburgersforbreakfast.ButIdon’tlikeeggs.IlikehamburgersButIdon’tlDoyourparentslikepizza?Yes,theydo.No,theydon’t.

Doyourparentslikepizza?Doesyourfatherlikebeer?Yes,hedoes.No,hedoesn't.Doesyourfatherlikebeer?Choosetherightwords.Therearefive_______(tomatoes,tomato)onthetable.The______(child,children)areintheroom.She_______(doesn’tlike,don’tlike)chickenforlunch.He______(likes,like)hamburgers.Choosetherightwords.5.Dothey_____(likes,like)apples?6.Mikedoesn’tlike______(bananas,banana).7.Mymotherdoesn’thave______(alunch,lunch)athome.8.Doesshe______(like,likes)Frenchfries?5.Dothey_____(likes,like)a用所给动词的适当形式填空。She______(notlike)salad._____you_____(like)oranges?He_____(eat)lotsofhealthyfoodeveryday.There_____(be)somechickenontheplate.Hismother_____(like)icecream.用所给动词的适当形式填空。6.______yourfather_____(like)eggsforbreakfast?7.They_____(notlike)playingtennis.8.There_____(be)somepotatoesinthebag.9.Mymother_____(have)supperathomeeveryday.10.Mr.Green_____(nothave)lunchatschooleveryday.6.______yourfather_____(likA:Doyouwanttogotoamovie?B:Yes,Ido.Thatsoundsgood.A:Whatkindofmoviesdoyoulike?B:Ilikeactionmoviesbecausetheyareexciting.AndIthinkcartoonsareinteresting.A:Howaboutthrillers?B:No.Thrillersarescary.Howaboutyou?A:Idon’tlikethrillers,either.ButIlikedocumentaries.Ithinkthey’reinstructive.A:DoyouwanttogotoamoviDoeshewanttogotoamovie?Yes,hedoes./No,hedoesn’t.Whatkindofmoviesdoeshewanttosee?Hewantstoseeacomedy.Doeshewanttogotoamovie?Whatdoyouliketodoonholidays?Whydoyoulike…?Whatdoyouliketodo二面角二面角二面角二面角

从空间一直线出发的两个半一、二面角的定义二、二面角的平面角角的平面角

一个平面垂直于二面角的棱，并与两半平面分别相交于射线PA、PB垂足为P，则∠APB叫做二面ABPγβαιαβι平面所组成的图形叫做二面角1、定义二面角二面角从空间一直线出发的两个半一、二面角的定义二、二面角的2、作二面角的平面角的常用方法①、点P在棱上②、点P在一个半平面上③、点P在二面角内ιpαβABABpαβιABOαβιp—定义法—三垂线定理法—垂面法二面角2、作二面角的平面角的常用方法①、点P在棱上②、点P在一个半1、如图，AB是圆的直径，PA垂直圆所在的平面，C是圆上任一点，则二面角P-BC-A的平面角为:A.∠ABPB.∠ACPC.都不是

练习2、已知P为二面角内一点，且P到两个半平面的距离都等于P到棱的距离的一半，则这个二面角的度数是多少？pαβιABOABCP60º二面角1、如图，AB是圆的直径，PA垂直圆所在的平面，C是圆上任一例1.如图，已知P是二面角α-AB-β棱上一点，过P分别在α、β内引射线PM、PN，且∠MPN=60º

∠BPM=∠BPN=45º

，求此二面角的度数。βαABPMNCDO解：在PB上取不同于P的一点O，在α内过O作OC⊥AB交PM于C，在β内作OD⊥AB交PN于D，连CD，可得∠COD是二面角α-AB-β的平面角设PO=a

，∵∠BPM=∠BPN=45º∴CO=a，

DO=a，

PCa，

PDa又∵∠MPN=60º

∴CD=PCa∴∠COD=90º因此，二面角的度数为90ºaOPC二面角例1.如图，已知P是二面角α-AB-β棱上一点，过P分别在α例2．如图P为二面角α–ι–β内一点，PA⊥α,PB⊥β,且PA=5，PB=8，AB=7，求这二面角的度数。

过PA、PB的平面PAB与棱ι

交于O点∵PA⊥α∴PA⊥ι

∵PB⊥β∴PB⊥ι

∴ι⊥平面PAB∴∠AOB为二面角α–ι–β的平面角又∵PA=5，PB=8，AB=7由余弦定理得∴∠P=60º∴∠AOB=120º

∴这二面角的度数为120º解：βαABPιO二面角例2．如图P为二面角α–ι–β内一点，PA⊥α,PB⊥β,且OABPC取AB的中点为E，连PE，OE∵O为AC中点，∠ABC=90º∴OE∥BC且

OEBC在Rt△POE中，OE

，PO∴∴所求的二面角P-AB-C的正切值为例3．如图，三棱锥P-ABC的顶点P在底面ABC上的射影是底面Rt△ABC斜边AC的中点O，若PB=AB=1，BC=，求二面角P-AB-C的正切值。∴∠PEO为二面角P-AB-C的平面角在Rt△PBE中，BE，PB=1，PEOE⊥AB，因此PE⊥ABE解：EOP二面角OABPC取AB的中点为E，连PE，OE∵O为AC中点练习1：已知Rt△ABC在平面α内，斜边AB在30º的二面角α-AB-β的棱上，若AC=5，BC=12，求点C到平面β的距离CO。βαACBOD练习2：在平面四边形ABCD中，AB=BC=2，AD=CD=,∠B=120º；将三角形ABC沿四边形ABCD的对角线AC折起来，使DB′=，求△AB′C所在平面与△ADC所在平面所成二面角的平面角的度数。ABCB’DO二面角练习1：已知Rt△ABC在平面α内，斜边AB在30º的二面角二、二面角的平面角一、二面角的定义

从空间一直线出发的两个半平面所组成的图形叫做二面角1、定义2、求二面角的平面角方法①点P在棱上②点P在一个半平面上③点P在二面角内ABPγβαι小结二面角二面角αβιABαβιpιpαβABpαβιABO—定义法—三垂线定理法—垂面法二面角二、二面角的平面角一、二面角的定义从空间一直线出发的αβABCD

A为二面角α–CD–β的棱CD上一点，AB在平面α内且与棱CD成45º角，又AB与平面β成30º，求二面角α–CD–β的大小。作业二面角二面角二面角αβABCDA为二面角α–CD–β的棱CD上一二面角谢谢！二面角谢谢！初一英语第二学期同步串讲第七讲

Doyouhaveaneraser?初一英语第二学期同步串讲Doyouhaveasoccerball?作二面角的平面角的常用方法课件soccerbasketballbadmintonbaseballballgamessoccerbasketballbadmintonbasetennis(racket)volleyballgolfPingpong(ball)tabletennistennis(racket)volleyballgolfPDoyouhaveanAmericanfootball/rugby?Yes,Ido./No,Idon’t.DoyouhaveanAmericanfootbaGolfclubDoyouhaveagolfball?Yes,Ido./No,Idon’t.GolfclubDoyouhaveagolfbaGroupworkNamesDoyouhavea…?BallsyouhaveLucyYes,Ido.basketballWhatballsdoyouhave?Ihave….GroupworkNamesDoyouhavea…1.你有一个排球吗？是的，我有。2.你有表兄妹吗?不,我没有。3.她有网球拍吗？是的，她有。7.咱们去踢球吧。Doyouhaveavolleyball?Yes,Ido.Doyouhaveanycousins?No,Idon’t.Doesshehaveatennisracket?Yes,shedoes.Let’splaysoccer.1.你有一个排球吗？是的，我有。Doyou4.他有英英字典吗？不，他没有。他有一本英汉字典。5.他们有笔记本吗？是的，他们有。6.让我看一看。DoeshehaveanEnglishEnglishdictionary?No,hedoesn’t.HehasanEnglishChinesedictionary?Dotheyhaveanynotebooks?Yes,theydo.Let’shavealook.4.他有英英字典吗？不，他没有。他有一本英汉字典。DoeDoeshehaveatennisracket?Yes,hedoes./No,hedoesn't.Doeshehaveatennisracket?YDoesshehaveasoccerball?Yes,shedoes./No,shedoesn't.Doesshehaveasoccerball?YeDoeshehaveabasketball?Yes,hedoes.Doeshehaveabasketball?Yes,Dotheyhaveapuck?Yes,theydo./No,theydon’t.icehockeyDotheyhaveapuck?icehockeyA:Iwanttoplaybaseball.Doyouhaveabaseballbat?B:No,Idon’t.A:Doesyourbrotherhaveone?B:Yes,hedoes.Let’sgoandfindhim.A:Iwanttoplaybaseball.DoLet’splayvolleyball.Thatsoundsgood.Let’splayvolleyball.Let’splaybarbell.BarbellThatsoundsboring.Let’splaybarbell.BarbellThatLet’splaybowling.Thatsoundsinteresting.Let’splaybowling.ThatsoundsLet’splaybaseball.Thatsoundsdifficult.Let’splaybaseball.Let’sgoskateboarding.Thatsoundsfun.(exciting)Let’sgoskateboarding.Let’splayPingPong.Thatsoundsrelaxing.letsb.dosth.Let’shaveabreak.Let’sgoandplayPing-pong.Let’splayPingPong.letsb.dOurSchoolSportsCenterneedsnewmembers.Fillintheenrollmentformifyouwanttojointheclubs.Firstname__________Lastname__________Age________________Sex________________Class______________Club_______________Reason__________________________________What’syourfirstname,please?Howoldareyou?Whatclassareyouin?Whatclubdoyouwanttojoin?Why?(Because…Ilike…)OurSchoolSportsCenterneed写出反义词dislike______interesting______good_______well________easy________lost_________day________different_______big________first_________on________infrontof_________this_______these_____________bring______写出反义词Fillintheblanks1.A:Excuseme,_____you____abaseball?B:No,I_____.A:______Jim_____one?B:Yes,he______.Let’sgoandfindhim.2.A:Hi,Sally.B:Hi,Jane.A:Sally,________myfriend,Lucy.B:Hi,Lucy._____tomeetyou.C:Niceto_____you,_____.dohavedon’tDoeshavedoesthisisNicemeettooFillintheblanksdoA:Let’splayvolleyball.B:That______interesting,butI_____haveone.A:Well,_____youhaveatennisbat?B:Yes,I_____.Let’sgoandplaytennis.A:That’sagood______.soundsdon’tdodoideaA:Let’splayvolleyball.soundDoesBhaveaneraser?Istheshopopen?Whatdayistoday?Whatcantheydo?No,shedoesn’t.No,itisn’t.Itisclosed.TodayisFriday.Theygotothesupermarket.Part1DoesBhaveaneraser?No,sheWhatwouldBlike?HowmuchmoneydoesBhave?Whereisherpurse?Whatdoesshedo?

Sheonlyhastwoyuan.Herpurseisathome.SheborrowsfouryuanfromC.Part2She’dlikearedpencilandapencil-sharpener.WhatwouldBlike?Part2She’dDoesJimhaveadictionary?DoesAnnhaveadictionary?WhohasanEnglish-Chinesedictionary?4.WhencanLiLeigiveitback?

Part3No,hedoesn’t.Lucyhasone.HecangiveitbacknextMonday.No,shedoesn’t.DoesJimhaveadictionary?P

close***closed

Theshopclosesat8pm.Pleaseclosethedoor.Thedoorisclosed.Theshopisclosed.Theshopisopen.close***closedThesh

borrow…from

Iborrowabookfromhim.MayIborrowapenfromyou?Don’tborrow

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thisafternoon?give…back(tosb.)PleThereissomethingwrongwith…Thereissomethingwrongwith

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Thereissomethingwrongwith…close***closedborrYouaregoingtohaveaP.Eclass,thinkaboutsomeactivities.Talkaboutthem.Time_________Place________Sport________Reason________YouaregoingtohaveaP.EclWritingWriteapassageaboutyourselfandyourfriend.Whatsportsdoyouplay?Whatsportsthingsdoyouhave?Whatotherthingsdoyoudo?Whataboutyourfriend?(parent)WritingWriteapassageaboutyKeypoints:

FoodDoyoulikebananas?Doeshe/shelikebananas?Doyoulikebananas?Keypoints:FoodDoyoulikebDoyoulikeoranges?Yes,Ido.No,Idon’t