阅读理解:如图1,已知AB//CD,∠PAB=130°,∠PCD=120°,求∠APC的度数.
小明的思路是:如图2、过点P作PE//AB,根据平行线的性质,可得∠APC=∠APE+∠CPE=110°,结合小明的解题思路,解决下列问题:
![](https://img.xkw.com/dksih/QBM/2022/4/11/2955837865254912/2962939379556352/STEM/6602e5421b03412a9f13330af906aa58.png?resizew=401)
(1)如图3,如果AB//CD∥EF,那么∠BAC+∠ACE+∠CEF的度数为 .
(2)如图4,已知AD∥BC,点P在射线OM上运动,当点P在A、B两点之间运动时,∠ADP=∠α,∠BCP=∠β,那么∠CPD,∠α,∠β之间有何数量关系?请说明理由;
(3)在(2)的条件下,如果点P在A、B两点外侧运动时(点P与点A、B、O三点不重合)请你直接写出∠CPD,∠α,∠β之间有何数量关系.
小明的思路是:如图2、过点P作PE//AB,根据平行线的性质,可得∠APC=∠APE+∠CPE=110°,结合小明的解题思路,解决下列问题:
![](https://img.xkw.com/dksih/QBM/2022/4/11/2955837865254912/2962939379556352/STEM/6602e5421b03412a9f13330af906aa58.png?resizew=401)
(1)如图3,如果AB//CD∥EF,那么∠BAC+∠ACE+∠CEF的度数为 .
(2)如图4,已知AD∥BC,点P在射线OM上运动,当点P在A、B两点之间运动时,∠ADP=∠α,∠BCP=∠β,那么∠CPD,∠α,∠β之间有何数量关系?请说明理由;
(3)在(2)的条件下,如果点P在A、B两点外侧运动时(点P与点A、B、O三点不重合)请你直接写出∠CPD,∠α,∠β之间有何数量关系.
更新时间:2022-04-21 14:18:30
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【推荐1】已知:如图,∠BAE+∠AED=180°,∠1=∠2,∠M和∠N有怎样的数量关系,并说明理由.
![](https://img.xkw.com/dksih/QBM/2020/5/23/2468767182839808/2469840387596288/STEM/fdf54a25cb99427bb614551e69b833f2.png?resizew=126)
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【推荐2】(1)如图1,若AB∥CD,则∠B+∠D=∠E,你能说明理由吗?
(2)反之,若∠B+∠D=∠E,直线AB与CD有什么位置关系?
(3)若将点E移至图2的位置,此时∠B,∠D,∠E之间有什么关系?
(4)若将点E移至图3的位置,此时∠B,∠D,∠E之间的关系又如何?
(5)在图4中,AB∥CD,∠E+∠G与∠B+∠F+∠D之间有何关系?
(2)反之,若∠B+∠D=∠E,直线AB与CD有什么位置关系?
(3)若将点E移至图2的位置,此时∠B,∠D,∠E之间有什么关系?
(4)若将点E移至图3的位置,此时∠B,∠D,∠E之间的关系又如何?
(5)在图4中,AB∥CD,∠E+∠G与∠B+∠F+∠D之间有何关系?
![](https://img.xkw.com/dksih/QBM/2017/9/12/1772518032588800/1773216093306880/STEM/7ea0f39e-1b94-4095-af1b-e9f46eb16532.png?resizew=512)
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【推荐3】如图1:在
中,
,(要求:点
在
上,点
在
上;![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04582116cd765fcc5a52f44279ad6c94.png)
(2)直角坐标系的建立,在代数和几何之间架起了一座桥梁,用代数的方法解决几何问题:某数学小组在自主学习时了解了三角形的中位线及相关的定理,在学习了相关知识后,该小组同学深入思考,利用中点坐标公式,给出了三角形中位线定理的另外一种证明方法.该数学小组建立如图2所示的直角坐标系,已知点
,
分别是
,
边的中点,不妨设点
,点
,
.请你利用该数学学习小组的思路证明
且
.(提示:中点坐标公式,
,
,
,
,则
,
中点坐标为
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30bf335d98f98e40e3d300487f1396ef.png)
(3)如图3:在
中,
,
,
,延长
至点
,
,连接
并延长
边于点
,若
,则
是否存在最小值,若存在求出最小值,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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(2)直角坐标系的建立,在代数和几何之间架起了一座桥梁,用代数的方法解决几何问题:某数学小组在自主学习时了解了三角形的中位线及相关的定理,在学习了相关知识后,该小组同学深入思考,利用中点坐标公式,给出了三角形中位线定理的另外一种证明方法.该数学小组建立如图2所示的直角坐标系,已知点
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【推荐1】【探索研究】已知:
和
都是等边三角形.
(1)如图
,若点
、
、
在一条直线上时,我们可以得到结论:线段
与
的数量关系为: ,线段
与
所成的锐角度数为
;
(2)如图2,当点
、
、
不在一条直线上时,
中的结论是否成立?如果成立,请证明;如果不成立,请说明理由;
【灵活运用】
(3)如图3,某广场是一个四边形区域
,现测得:
,
,且
,
,试求圆形水池两旁
、
两点之间的距离.
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(1)如图
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(2)如图2,当点
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【灵活运用】
(3)如图3,某广场是一个四边形区域
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【推荐2】定义:如果三角形的两个内角
与
满足
,那么称这样的三角形为“微妙三角形”,从三角形一个顶点引出一条射线与对边相交,顶点与交点之间的线段把这个三角形分割成两个小三角形,如果分得的两个小三角形中有一个是“微妙三角形”,我们就把这条线段叫做这个三角形的“微妙分割线”.
理解概念:
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/8/aaa309e3-2ea0-483a-a666-fcc88a69f3c9.png?resizew=265)
(1)如图①,在
中,
,
平分
.求证:
为“微妙三角形”;
概念应用:
(2)若
为“微妙三角形”,且
.试判断
的形状,并说明理由;
(3)如图②,在
中,若
,
平分
,且
是
的“微妙分割线”,请直接写出
的度数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67a772a22816722cf3d69e596332b719.png)
理解概念:
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/8/aaa309e3-2ea0-483a-a666-fcc88a69f3c9.png?resizew=265)
(1)如图①,在
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deece9a4d9301fb6028aa1f81a11c23b.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
概念应用:
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deece9a4d9301fb6028aa1f81a11c23b.png)
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(3)如图②,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65c5840e0454fdec7d2158072c43b8db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
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