这部分需要会什么？
- 使用对顶角、邻补角和平行线角关系
- 求三角形与多边形内角
- 把图中的角关系转为方程
核心概念与公式
对顶角相等，组成平角的邻角之和为 \(180^\circ\)，围绕一点的角之和为 \(360^\circ\)。两条平行线被截时，同位角和内错角相等，同旁内角互补。
三角形内角和为 \(180^\circ\)，一个外角等于不相邻两内角之和。\(n\) 边形内角和为 \((n-2)180^\circ\)；正 \(n\) 边形每个内角为该总和除以 \(n\)，每个外角为 \(360^\circ/n\)。
做题时怎样使用
- 先标出已知平行、垂直、等腰等条件
- 用最直接的角关系建立等式
- 算出角后检查它属于内角、外角还是补角，并确认大小合理
容易混淆的地方
只有给出或证明平行时，才能使用平行线的对应角关系；不能仅凭图看起来平行。多边形内角相等也需要正多边形等条件。
用一个例题串起方法
知识点教学例题三角形一个外角为 \(125^\circ\)，不相邻的两个内角分别为 \((2x+5)^\circ\) 和 \(3x^\circ\)。求 \(x\)。
展开解题思路 ＋
外角等于两不相邻内角之和，因此 \(2x+5+3x=125\)，得到 \(x=24\)。两个内角为 \(53^\circ\)、\(72^\circ\)，第三个为 \(55^\circ\)。
本知识点题库
先独立作答，再查看答案与解析。
直线・角与多边形挑战
In the figure, line m is parallel to line n, line t intersects both lines, and \( 180<x+y+z<242 \) . Which of the following could NOT be a value of x?
I. 6
II. 61
III. 62
参考答案
直线・角与多边形进阶
In triangle ABC, the measure of angle B is 42° and the measure of angle C is 24°. What is the measure of angle A?
参考答案
直线・角与多边形挑战
In the figure shown, line r is parallel to line s, and both lines are intersected by line t. If \( x=2w+21 \), \( y=6w+28 \), and \( 8w=a \), where a is a constant, what is the value of a?
参考答案
直线・角与多边形基础
In the figure, line u is parallel to line v, and line t intersects both lines. What is the value of x?
参考答案
直线・角与多边形进阶
In the figure, line p is parallel to line q, and line t intersects both lines. What is the value of \( x+162 \) ?
参考答案
直线・角与多边形基础
In triangle ABC, the measure of angle B is 39° and the measure of angle C is 33°. What is the measure of angle A?
参考答案
直线・角与多边形挑战
In the figure, line q is parallel to line r, and both lines are intersected by line s. If \( y=2x+7 \), what is the value of x?
参考答案
直线・角与多边形进阶
In triangle ABC, the measure of angle B is 49° and the measure of angle C is 27°. What is the measure of angle A?
参考答案
直线・角与多边形基础
In the figure, line p is parallel to line q, and line t intersects both lines. What is the value of \( x+149 \)?
参考答案
直线・角与多边形挑战
In triangle XYZ, the measure of angle X is \( x^{∘} \), the measure of angle Y is \( y^{∘} \), and the measure of angle Z is 32°. If \( x>y>32 \), which of the following additional pieces of information is NOT sufficient to determine the values of x and y?
参考答案
直线・角与多边形进阶
In △JKL , the measures of both ∠J and ∠K are equal and the measure of ∠L is 136°. What is the measure of ∠J?
原库参考解析
直线・角与多边形挑战
In the figure shown, point T lies on segment SW, and segment TU is parallel to segment VW. The measure of angle V is 26°, and the measure of angle STU is 74°. What is the value of \( p−r \) ?
原库参考解析
