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 \( RST \) , the measure of angle \( R \) is 50°, the measure of angle \( S \) is x°, and the measure of angle \( T \) is \( \left(4x-5\right)° \). Point \( L \) lies on \( RS \) , point \( K \) lies on \( ST \) and \( LK \) is parallel to \( RT \) . What is the measure, in degrees, of angle \( SKL \) ? (Disregard the degree symbol when entering your answer.)
原库参考解析
直线・角与多边形进阶
In \( \triangle DEF \) , the measure of \( ∠D \) is 114°. Which of the following could be the measure, in degrees, of \( ∠E \) ?
原库参考解析
直线・角与多边形挑战
In the figure, line l is parallel to line m. What is the value of x?
原库参考解析
直线・角与多边形进阶
In triangle ABC, the measure of angle A is 34° and the measure of angle B is 90°. 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 shown, line l intersects lines j and k. Which additional piece of information is sufficient to prove that lines j and k are parallel?
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 \( \left(7x-260\right)° \). 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
A circle has center \( O \), and points \( R \) and \( S \) lie on the circle. In triangle \( ORS \), the measure of \( ∠ROS \) is 116°. What is the measure of \( ∠RSO \), in degrees? (Disregard the degree symbol when entering your answer.)
原库参考解析
直线・角与多边形进阶
In the figure shown, lines \( l \) and \( k \) are parallel and line \( t \) intersects both lines. If \( x>145 \), which of the following must be true?
原库参考解析
直线・角与多边形进阶
In triangle FGH, the measure of angle F is 63°. Which additional piece of information is sufficient to prove that triangle FGH is a right triangle?
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. 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, 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
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 \( (7x−410)° \). The sum of the measures of one of the acute angles and three of the obtuse angles is \( (−14x+w)° \). What is the value of w?
原库参考解析
