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 the figure shown, 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
A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of the acute angles is \( (9x−650)^{∘} \) . The sum of the measures of one of the acute angles and three of the obtuse angles is \( (−18x+w)^{∘} \). What is the value of \( w \)?
原库参考解析
直线・角与多边形进阶
In the figure shown, the length of \( AB \) is 48. Which statement provides sufficient information to prove that triangle \( ABC \) is isosceles?
原库参考解析
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?
原库参考解析
直线・角与多边形挑战
A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of the acute angles is (9x-650)°. The sum of the measures of one of the acute angles and three of the obtuse angles is (-18x+w)° . What is the value of w?
原库参考解析
直线・角与多边形挑战
In triangle EFG, the sum of the measures of angles E and F is 31°. What is the measure of angle G?
原库参考解析
直线・角与多边形进阶
In the figure shown, line r is parallel to line s, and line t intersects both lines. What is the value of x ?
原库参考解析
直线・角与多边形进阶
In the figure shown, line r is parallel to line s, and line t intersects both lines. What is the value of x?
原库参考解析
直线・角与多边形进阶
In triangle \( PQR \), the measure of \( ∠P \) is 56°, and the measure of \( ∠Q \) is 74°. What is the measure of \( ∠R \), in degrees?
原库参考解析
直线・角与多边形进阶
In the figure, lines r and s are parallel and line t intersects both lines. If \( a=4k+57 \) and \( b=\frac{k}{4}+55 \), what is the value of c?
原库参考解析
直线・角与多边形进阶
In the figure, line p is parallel to line r, 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
In the figure shown, line k intersects lines r and s. If \( w=147 \), which additional piece of information is sufficient to prove that lines r and s are parallel?
原库参考解析
