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INCLUDEPICTURE "节.tif" INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/节.tif" \* MERGEFORMAT INCLUDEPICTURE "../../节.tif" \* MERGEFORMAT INCLUDEPICTURE "../../节.tif" \* MERGEFORMAT INCLUDEPICTURE "../节.tif" \* MERGEFORMAT INCLUDEPICTURE "节.tif" \* MERGEFORMAT
6.3　角
6．3.1　角的概念
eq \o(\s\up7( INCLUDEPICTURE "知识梳理.tif" INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/知识梳理.tif" \* MERGEFORMAT INCLUDEPICTURE "../../知识梳理.tif" \* MERGEFORMAT INCLUDEPICTURE "../../知识梳理.tif" \* MERGEFORMAT INCLUDEPICTURE "../知识梳理.tif" \* MERGEFORMAT INCLUDEPICTURE "知识梳理.tif" \* MERGEFORMAT ),\s\do5(　　　　　　　))
知识点一　角的认识
1．有公共端点的两条射线组成的图形叫作角，这个公共端点是角的顶点，这两条射线是角的两条边．
2．角也可以看作由一条射线绕着它的端点旋转而形成的图形．
练习1　下列图形中，能用∠O和∠1表示同一个角的是(　C　).
eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX186.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX186.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX186.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX186.tif" \* MERGEFORMAT INCLUDEPICTURE "SX186.tif" \* MERGEFORMAT ),\s\do5(A)) 　　 eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX186A.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX186A.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX186A.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX186A.tif" \* MERGEFORMAT INCLUDEPICTURE "SX186A.tif" \* MERGEFORMAT ),\s\do5(B))
eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX186b.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX186b.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX186b.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX186b.tif" \* MERGEFORMAT INCLUDEPICTURE "SX186b.tif" \* MERGEFORMAT ),\s\do5(C)) 　　 eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX186C.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX186C.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX186C.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX186C.tif" \* MERGEFORMAT INCLUDEPICTURE "SX186C.tif" \* MERGEFORMAT ),\s\do5(D))
【解析】要是能用∠1，∠O表示同一个角，必须共用角的顶点，且角的两边重合．选项A，B，D中，以O为顶点的角不止一个，用∠O表示不明确，不符合题意；选项C符合题意，故选C.
总结　角的表示：①用三个大写字母可以表示任一角(顶点字母在中间)；②用一个顶点字母表示角的前提是以该点作为顶点的角只有一个；③用数字或希腊字母表示角时，要在靠近角的顶点处画上弧线，并标上数字或希腊字母．
知识点二　角的单位与换算
3．以度、分、秒为单位的角的度量制，叫作角度制．
4．1周角＝360°＝2平角，1平角＝180°＝2直角，1直角＝90°，1°＝60′，1′＝60″.
练习2　把12°15′36″用度表示正确的是(　B　).
A．12.11° B．12.26°
C．12.16° D.11.26°
【解析】因为1′＝60″，
所以36″＝0.6′.
因为1°＝60′，
所以15.6′＝0.26°，
所以12°15′36″＝12.26°.故选B.
总结　在进行单位换算时，把高级单位转化为低级单位，要乘进率，把低级单位转化为高级单位，要除以进率．
知识点三　方向角
5．方向角通常用南偏东多少度，南偏西多少度，北偏西多少度，北偏东多少度来表示．
练习3　如图，下列说法错误的是(　B　).
INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX187.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX187.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX187.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX187.tif" \* MERGEFORMAT INCLUDEPICTURE "SX187.tif" \* MERGEFORMAT
A．OB的方向是北偏西15°
B．OA的方向是北偏东30°
C．OC的方向是南偏西25°
D．OD的方向是东南方向
【解析】A.OB的方向是北偏西15°，此选项正确，不符合题意；B.OA的方向是北偏东60°，此选项错误，符合题意；C.OC的方向是南偏西25°，此选项正确，不符合题意；D.OD的方向是东南方向，此选项正确，不符合题意．故选B.
总结　在表示方向角时，一般以正南或正北方向为基准，再加上偏东或偏西的角度．
eq \o(\s\up7( INCLUDEPICTURE "课后练习.tif" INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/课后练习.tif" \* MERGEFORMAT INCLUDEPICTURE "../../课后练习.tif" \* MERGEFORMAT INCLUDEPICTURE "../../课后练习.tif" \* MERGEFORMAT INCLUDEPICTURE "../课后练习.tif" \* MERGEFORMAT INCLUDEPICTURE "课后练习.tif" \* MERGEFORMAT ),\s\do5(　　　　　　　))
INCLUDEPICTURE "三角.tif" INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/三角.tif" \* MERGEFORMAT INCLUDEPICTURE "../../三角.tif" \* MERGEFORMAT INCLUDEPICTURE "../../三角.tif" \* MERGEFORMAT INCLUDEPICTURE "../三角.tif" \* MERGEFORMAT INCLUDEPICTURE "三角.tif" \* MERGEFORMAT 基础巩固
1．图中能用一个大写字母表示的角有(　C　).
INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX189.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX189.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX189.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX189.tif" \* MERGEFORMAT INCLUDEPICTURE "SX189.tif" \* MERGEFORMAT
A．4个 B．3个
C．2个 D．1个
2．如图所示，点O是直线AB上一点，图中小于180°的角共有(　B　).
INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/X+1.tif" \* MERGEFORMAT INCLUDEPICTURE "../../X+1.tif" \* MERGEFORMAT INCLUDEPICTURE "../../X+1.tif" \* MERGEFORMAT INCLUDEPICTURE "../X+1.tif" \* MERGEFORMAT INCLUDEPICTURE "X+1.tif" \* MERGEFORMAT
A．10个 B．9个
C．8个 D.7个
【解析】题图中小于180°的角有∠AOC，∠AOD，∠AOE，∠COD，∠COE，∠COB，∠DOE，∠DOB，∠EOB，共9个．故选B.
3．把20°13′12″用度表示正确的是(　A　).
A.20.22° B．20.02°
C．20.2° D．20.12°
【解析】因为1′＝60″，所以12″＝0.2′，因为1°＝60′，所以13.2′＝0.22°，
所以20°13′12″＝20.22°.故选A.
4．下列图形中，能用∠1，∠ACB，∠C三种方法表示同一个角的是(　A　).
eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX191.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX191.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX191.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX191.tif" \* MERGEFORMAT INCLUDEPICTURE "SX191.tif" \* MERGEFORMAT ),\s\do5( A)) 　　 eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX191A.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX191A.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX191A.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX191A.tif" \* MERGEFORMAT INCLUDEPICTURE "SX191A.tif" \* MERGEFORMAT ),\s\do5(B))
eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX191b.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX191b.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX191b.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX191b.tif" \* MERGEFORMAT INCLUDEPICTURE "SX191b.tif" \* MERGEFORMAT ),\s\do5(C　)) eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX191C.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX191C.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX191C.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX191C.tif" \* MERGEFORMAT INCLUDEPICTURE "SX191C.tif" \* MERGEFORMAT ),\s\do5( D))
5．如图，由点A观测点B的方向是______________，由点B观测点A的方向是______________．
INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/24S60.tif" \* MERGEFORMAT INCLUDEPICTURE "../../24S60.tif" \* MERGEFORMAT INCLUDEPICTURE "../../24S60.tif" \* MERGEFORMAT INCLUDEPICTURE "../24S60.tif" \* MERGEFORMAT INCLUDEPICTURE "24S60.tif" \* MERGEFORMAT
【答案】南偏东60°　北偏西60°
INCLUDEPICTURE "三角.tif" INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/三角.tif" \* MERGEFORMAT INCLUDEPICTURE "../../三角.tif" \* MERGEFORMAT INCLUDEPICTURE "../../三角.tif" \* MERGEFORMAT INCLUDEPICTURE "../三角.tif" \* MERGEFORMAT INCLUDEPICTURE "三角.tif" \* MERGEFORMAT 能力达标
6．如图，用量角器测得∠ABC的度数是(　C　).
INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX196.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX196.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX196.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX196.tif" \* MERGEFORMAT INCLUDEPICTURE "SX196.tif" \* MERGEFORMAT
A．50° B．80°
C．130° D．150°
7．如图，在锐角∠AOB内部，画1条射线，可以得到3个锐角；画2条不同的射线，可以得到6个锐角；画3条不同的射线，可以得到10个锐角；…….照此规律，画19条不同的射线，可以得到锐角的个数为(　A　).
eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX194.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX194.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX194.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX194.tif" \* MERGEFORMAT INCLUDEPICTURE "SX194.tif" \* MERGEFORMAT ),\s\do5(（1）)) 　 eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX194A.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX194A.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX194A.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX194A.tif" \* MERGEFORMAT INCLUDEPICTURE "SX194A.tif" \* MERGEFORMAT ),\s\do5(（2）)) 　 eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX194B.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX194B.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX194B.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX194B.tif" \* MERGEFORMAT INCLUDEPICTURE "SX194B.tif" \* MERGEFORMAT ),\s\do5(（3）)) 　
A．210 B．199
C．186 D．165
【解析】第(1)个图中有1＋2＝3(个)锐角，第(2)个图中有1＋2＋3＝6(个)锐角，第(3)个图中有1＋2＋3＋4＝10(个)锐角，……
按题图所示的规律画n条不同的射线，可以得到锐角的个数为1＋2＋3＋…＋(n＋1)＝，
所以画19条不同的射线，可以得到锐角的个数为＝210.
故选A.
8．某市场的公平秤如图所示，把10 kg的菜放到秤上，指示盘上的指针转了180°.
INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX300.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX300.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX300.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX300.tif" \* MERGEFORMAT INCLUDEPICTURE "SX300.tif" \* MERGEFORMAT
(1)如果把3 kg的菜放到秤上，指针转了多少度？
(2)如果称好0.5 kg的菜没有拿走，再把一捆菜放到秤上，指针共转了72°54′，那么后放上的这捆菜重多少千克？
【解】(1)设把3 kg的菜放到秤上，指针转了x°.
根据题意，得＝，
解得x＝54.
答：指针转了54°.
(2)72°54′＝72.9°.
设后放上的这捆菜重y kg.
根据题意，得＝，
解得y＝3.55.
答：后放上的这捆菜重3.55 kg.
INCLUDEPICTURE "三角.tif" INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/三角.tif" \* MERGEFORMAT INCLUDEPICTURE "../../三角.tif" \* MERGEFORMAT INCLUDEPICTURE "../../三角.tif" \* MERGEFORMAT INCLUDEPICTURE "../三角.tif" \* MERGEFORMAT INCLUDEPICTURE "三角.tif" \* MERGEFORMAT 挑战创新
9．钟面上角的问题．
(1)3点45分，时针与分针的夹角是多少？
(2)在9点与10点之间，什么时候时针与分针成100°的角？
【解】(1)如图1所示，
因为由3点到3点45分，分针转了270°，时针转了270°×，
所以时针与分针的夹角是
180°－270°×＝157.5°.
(2)设分针转的度数为x°，则时针转的度数为.
由图2，得90＋x－＝100，
解得x＝.
÷6＝(分).
由图3，得90＋－(x－180)＝100，
解得x＝.
÷6＝(分).
所以9点过或分钟时，时针与分针成100°的角．
eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX335A.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX335A.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX335A.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX335A.tif" \* MERGEFORMAT INCLUDEPICTURE "SX335A.tif" \* MERGEFORMAT ),\s\do5(图1)) 　 eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX335B.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX335B.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX335B.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX335B.tif" \* MERGEFORMAT INCLUDEPICTURE "SX335B.tif" \* MERGEFORMAT ),\s\do5(图2)) 　 eq \o(\s\up7( INCLUDEPICTURE "../../../../广东慧源同步上册/广东慧源七年级上册数学/SX335C.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX335C.tif" \* MERGEFORMAT INCLUDEPICTURE "../../SX335C.tif" \* MERGEFORMAT INCLUDEPICTURE "../SX335C.tif" \* MERGEFORMAT INCLUDEPICTURE "SX335C.tif" \* MERGEFORMAT ),\s\do5(图3))6.3　角
6．3.1　角的概念
eq \o(\s\up7(),\s\do5(　　　　　　　))
知识点一　角的认识
1．有公共端点的两条射线组成的图形叫作角，这个公共端点是角的顶点，这两条射线是角的两条边．
2．角也可以看作由一条射线绕着它的端点旋转而形成的图形．
练习1　下列图形中，能用∠O和∠1表示同一个角的是(　).
eq \o(\s\up7(),\s\do5( )) 　　 eq \o(\s\up7(),\s\do5( ))
eq \o(\s\up7(),\s\do5( )) 　　 eq \o(\s\up7(),\s\do5( ))
知识点二　角的单位与换算
3．以度、分、秒为单位的角的度量制，叫作角度制．
4．1周角＝360°＝2平角，1平角＝180°＝2直角，1直角＝90°，1°＝60′，1′＝60″.
练习2　把12°15′36″用度表示正确的是(　).
A．12.11° B．12.26°
C．12.16° D.11.26°
知识点三　方向角
5．方向角通常用南偏东多少度，南偏西多少度，北偏西多少度，北偏东多少度来表示．
练习3　如图，下列说法错误的是(　).
A．OB的方向是北偏西15°
B．OA的方向是北偏东30°
C．OC的方向是南偏西25°
D．OD的方向是东南方向
eq \o(\s\up7(),\s\do5(　　　　　　　))
基础巩固
1．图中能用一个大写字母表示的角有(　).
A．4个 B．3个
C．2个 D．1个
2．如图所示，点O是直线AB上一点，图中小于180°的角共有(　).
A．10个 B．9个
C．8个 D.7个
3．把20°13′12″用度表示正确的是(　).
A.20.22° B．20.02°
C．20.2° D．20.12°
4．下列图形中，能用∠1，∠ACB，∠C三种方法表示同一个角的是(　).
eq \o(\s\up7(),\s\do5( A)) 　　 eq \o(\s\up7(),\s\do5( ))
eq \o(\s\up7(),\s\do5(C　)) eq \o(\s\up7(),\s\do5( D))
5．如图，由点A观测点B的方向是______________，由点B观测点A的方向是______________．
能力达标
6．如图，用量角器测得∠ABC的度数是(　).
A．50° B．80°
C．130° D．150°
7．如图，在锐角∠AOB内部，画1条射线，可以得到3个锐角；画2条不同的射线，可以得到6个锐角；画3条不同的射线，可以得到10个锐角；…….照此规律，画19条不同的射线，可以得到锐角的个数为(　).
eq \o(\s\up7(),\s\do5(（1）)) 　 eq \o(\s\up7(),\s\do5(（2）)) 　 eq \o(\s\up7(),\s\do5(（3）)) 　
A．210 B．199
C．186 D．165
8．某市场的公平秤如图所示，把10 kg的菜放到秤上，指示盘上的指针转了180°.
(1)如果把3 kg的菜放到秤上，指针转了多少度？
(2)如果称好0.5 kg的菜没有拿走，再把一捆菜放到秤上，指针共转了72°54′，那么后放上的这捆菜重多少千克？
挑战创新
9．钟面上角的问题．
(1)3点45分，时针与分针的夹角是多少？
(2)在9点与10点之间，什么时候时针与分针成100°的角？

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