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7．2.3　平行线的性质
1.掌握平行线的性质定理Ⅰ：两条平行直线被第三条直线所截，同位角相等．2．探索并证明平行线的性质定理Ⅱ：两条平行直线被第三条直线所截，内错角相等(或同旁内角互补).3．能用平行线的性质进行简单的推理和计算．4．能熟练地运用平行线的判定与性质进行推理和计算．
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知识点一　平行线的性质
1．性质1：两条平行直线被第三条直线所截，同位角 ．简单说成：两直线平行，同位角 ．
2．性质2：两条平行直线被第三条直线所截，内错角 ．简单说成：两直线平行，内错角 ．
3．性质3：两条平行直线被第三条直线所截，同旁内角 ．简单说成：两直线平行，同旁内角 ．
练习1　如图，已知AB∥DE，BC∥EF，求∠B＋∠E的度数．
1．如图，平行线AB，CD被直线EF所截，FG平分∠EFD.若∠EFD＝78°，则∠EGF的度数是( ).
A．39° B．51°
C．78° D．102°
2．如图，一块直角三角板的直角顶点放在直尺的一边上．如果∠1＝70°，那么∠2的度数是( ).
A．20° B．25°
C．30° D．45°
3．如图，直线a∥b，直线l与a，b分别交于点A，B，过点A作AC⊥b于点C，若∠1＝60°，则∠2的度数为( ).
A．30° B．45°
C．60° D．120°
4．如图，直尺一边BC与量角器的零刻度线AD平行，已知∠EOD的读数为65°，设OE与BC交于点F，则∠BFE的度数等于( ).
A．125° B．115°
C．105° D．95°
5．如图，已知∠1＋∠2＝180°，∠3＝∠B，试说明DE∥BC.
6．如图，自行车的尾部通常会安装一种塑料制成的反光镜，夜间骑车时，在车灯照射下，能把光线按原来方向返回(即a∥b)，根据光的反射可知∠1＝∠3，∠2＝∠4，若∠1＝46°，求∠2的度数．
7．如图，已知AB∥CD，∠ABE＝150°，∠CDE＝85°，求∠BED的度数．
8．某消防云梯示意图如图1所示，其由救援台AB、延展臂BC(B在C的左侧)、伸展主臂CD、支撑臂EF构成．在作业过程中，救援台AB、车身GH及地面MN三者始终保持水平平行．为了参与一项高空救援工作，需要进行作业调整，如图2，使得延展臂BC与支撑臂EF所在直线互相垂直，且∠EFH＝71°，则这时展角∠ABC＝____________．
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1.掌握平行线的性质定理Ⅰ：两条平行直线被第三条直线所截，同位角相等．2．探索并证明平行线的性质定理Ⅱ：两条平行直线被第三条直线所截，内错角相等(或同旁内角互补).3．能用平行线的性质进行简单的推理和计算．4．能熟练地运用平行线的判定与性质进行推理和计算．
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知识点一　平行线的性质
1．性质1：两条平行直线被第三条直线所截，同位角相等．简单说成：两直线平行，同位角相等．
2．性质2：两条平行直线被第三条直线所截，内错角相等．简单说成：两直线平行，内错角相等．
3．性质3：两条平行直线被第三条直线所截，同旁内角互补．简单说成：两直线平行，同旁内角互补．
练习1　如图，已知AB∥DE，BC∥EF，求∠B＋∠E的度数．
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【答案】180°
知识点二　利用平行线的性质解决折叠问题
练习2　如图，将一张长方形纸片(其中AD∥BC)沿EF折叠后，使得点A，B分别落在点A′，B′的位置．若∠2＝60°，求∠1的度数．
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【解】因为AD∥BC，
所以∠B′FC＝∠2＝60°.
由折叠的性质可知∠1＝∠B′FE，
又∠1＋∠B′FE＋∠B′FC＝180°，
所以∠1＝∠B′FE＝(180°－∠B′FC)＝×(180°－60°)＝60°.
知识点三　利用平行线的性质解决实际问题
练习3　如图，一艘船在海面上航行，到达B处时，看到灯塔A在它的北偏东40°方向，达到C处时，看到灯塔A在它的北偏西30°方向，则∠BAC＝70°．
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知识点四　平行线的判定和性质的综合应用
练习4　(教材P17例3变式)将下面的解答过程补充完整：如图，已知DE∥BC，EF平分∠CED，∠A＝∠CFE，那么EF与AB平行吗？为什么？
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解：EF∥AB.理由如下：
因为DE∥BC(已知)，
所以∠DEF＝∠CFE(两直线平行，内错角相等).
因为EF平分∠CED(已知)，
所以∠DEF＝∠CEF(角平分线的定义)，
所以∠CFE＝∠CEF(等量代换).
因为∠A＝∠CFE(已知)，
所以∠A＝∠CEF(等量代换)，
所以EF∥AB(同位角相等，两直线平行).
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1．如图，平行线AB，CD被直线EF所截，FG平分∠EFD.若∠EFD＝78°，则∠EGF的度数是(　A　).
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A．39° B．51°
C．78° D．102°
2．如图，一块直角三角板的直角顶点放在直尺的一边上．如果∠1＝70°，那么∠2的度数是(　A　).
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A．20° B．25°
C．30° D．45°
3．如图，直线a∥b，直线l与a，b分别交于点A，B，过点A作AC⊥b于点C，若∠1＝60°，则∠2的度数为(　A　).
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A．30° B．45°
C．60° D．120°
4．如图，直尺一边BC与量角器的零刻度线AD平行，已知∠EOD的读数为65°，设OE与BC交于点F，则∠BFE的度数等于(　B　).
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A．125° B．115°
C．105° D．95°
5．如图，已知∠1＋∠2＝180°，∠3＝∠B，试说明DE∥BC.
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【解】因为∠1＋∠2＝180°，∠1＋∠DFE＝180°，
所以∠2＝∠DFE，
所以AB∥EF，
所以∠3＝∠ADE.
因为∠3＝∠B，
所以∠ADE＝∠B，
所以DE∥BC.
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6．如图，自行车的尾部通常会安装一种塑料制成的反光镜，夜间骑车时，在车灯照射下，能把光线按原来方向返回(即a∥b)，根据光的反射可知∠1＝∠3，∠2＝∠4，若∠1＝46°，求∠2的度数．
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【解】如图，
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因为∠1＝∠3＝46°，
所以∠MAB＝180°－46°－46°＝88°.
因为a∥b，
所以∠ABN＋∠MAB＝180°.
因为∠2＋∠4＋∠ABN＝180°，
所以∠2＋∠4＝∠MAB.
因为∠2＝∠4，
所以∠2＝×88°＝44°.
7．如图，已知AB∥CD，∠ABE＝150°，∠CDE＝85°，求∠BED的度数．
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【答案】55°
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8．某消防云梯示意图如图1所示，其由救援台AB、延展臂BC(B在C的左侧)、伸展主臂CD、支撑臂EF构成．在作业过程中，救援台AB、车身GH及地面MN三者始终保持水平平行．为了参与一项高空救援工作，需要进行作业调整，如图2，使得延展臂BC与支撑臂EF所在直线互相垂直，且∠EFH＝71°，则这时展角∠ABC＝____________．
eq \o(\s\up7( INCLUDEPICTURE "L26CGD7SX22A.TIF" INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22A.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22A.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22A.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22A.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22A.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22A.TIF" \* MERGEFORMATINET ),\s\do5(图1))
eq \o(\s\up7( INCLUDEPICTURE "L26CGD7SX22B.TIF" INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22B.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22B.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22B.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22B.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22B.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\L26CGD7SX22B.TIF" \* MERGEFORMATINET ),\s\do5(图2))
【答案】161°

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