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・角与多边形进阶
In triangle \( ABC \), the sum of the measure of angle \( A \) and the measure of angle \( B \) is 118 degrees. What is the measure, in degrees, of angle \( C \) ?
原库参考解析
直线・角与多边形挑战
In the figure, line n is parallel to line s, and both lines are intersected by line t. If \( x=6z-87 \) and \( y=3z+15 \), what is the value of z ?
原库参考解析
直线・角与多边形基础
In the figure, line m is parallel to line n, and line t intersects both lines. What is the value of x?
原库参考解析
直线・角与多边形挑战
Triangles \( PQR \) and \( LMN \) are graphed in the xy-plane. Triangle \( PQR \) has vertices \( P \), \( Q \), and \( R \) at \( \left(4,\ 5\right) \), \( \left(4,\ 7\right) \), and \( \left(6,\ 5\right) \), respectively. Triangle \( LMN \) has vertices \( L \), \( M \), and \( N \) at \( \left(4,\ 5\right) \), \( \left(4,\ 7+k\right) \) , and \( \left(6+k,\ 5\right) \), respectively, where k is a positive constant. If the measure of ∠\( Q \) is t°, what is the measure of ∠\( N \)?
原库参考解析
直线・角与多边形基础
In the triangle shown, what is the value of x?
原库参考解析
直线・角与多边形基础
In the figure shown, \( BC \) is parallel to \( AD \) and \( x=22 \). Line \( t \) passes through points \( A \) and \( C \). What is the value of \( y \)?
原库参考解析
直线・角与多边形挑战
Two lines intersect at exactly one point, forming two acute angles and two obtuse angles. The measure of one of these angles is \( \left(9x-140\right)° \). Which of the following could NOT be the sum of the measures of any two of these angles?
原库参考解析
直线・角与多边形进阶
In triangle ABC, the measure of angle B is 52° and the measure of angle C is 17°. What is the measure of angle A?
原库参考解析
直线・角与多边形基础
In the figure, line j is parallel to line k. What is the value of x?
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. What is the value of w?
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・角与多边形挑战
In the figure, line m is parallel to line n, and line t intersects both lines. What is the value of x?
原库参考解析
