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・角与多边形基础
In the figure shown, line m is parallel to line n. What is the value of x?
原库参考解析
直线・角与多边形挑战
In the figure, lines m and n are parallel.If x = 6k + 13 and y= 8k - 29 what is the value of z?
原库参考解析
直线・角与多边形进阶
In the figure, line m is parallel to line n, and line k intersects both lines. Which of the following statements is true?
原库参考解析
直线・角与多边形挑战
In \( \triangle RST \) , the measure of ∠R is 63°. Which of the following could be the measure, in degrees, of ∠S?
原库参考解析
直线・角与多边形基础
In the figure, two lines intersect at a point. Angle 1 and angle 2 are vertical angles. The measure of angle 1 is 72° . What is the measure of angle 2?
原库参考解析
直线・角与多边形基础
In the figure, line n intersects lines r and s. Line r is parallel to line s. What is the value of x?
原库参考解析
直线・角与多边形挑战
In triangle \( XYZ \), angle \( Y \) is a right angle, point \( P \) lies on \( XZ \) , and point \( Q \) lies on \( Y Z \) such that \( PQ \) is parallel to \( XY \) . If the measure of angle \( XZY \) is 63°, what is the measure, in degrees, of angle \( XPQ \)?
原库参考解析
直线・角与多边形进阶
In triangle ABC, the measure of angle A is 54°, the measure of angle B is 90°, and the measure of angle C is \( (\frac{k}{2})^{°} \) . What is the value of k?
原库参考解析
直线・角与多边形基础
In the figure shown, line c intersects parallel lines s and t. What is the value of x?
原库参考解析
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 \( x+142 \)?
原库参考解析
直线・角与多边形挑战
In the figure above, RT = TU. What is the value of x ?
原库参考解析
直线・角与多边形挑战
In the figure, parallel lines q and t are intersected by lines r and s. If a = 43 and b = 122, what is the value of w ?
原库参考解析
