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 measure of angle \( A \) is 53°, the measure of angle \( B \) is 90°, and the measure of angle \( C \) is \( \left(\frac{k}{2}\right)° \). What is the value of k?
原库参考解析
直线・角与多边形进阶
In the figure, two lines intersect at a point. If \( r=61 \) , what is the value of s?
原库参考解析
直线・角与多边形挑战
A polygon has exactly 87 sides. If the measure of each of the 87 interior angles of this polygon is (180p)°, what is the value of p?
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 \( x>154 \) , which of the following must be true?
原库参考解析
直线・角与多边形挑战
For triangle \(ABC\), the length of \(AB\) is 11 less than the length of \(AC\). Point \(D\) (not shown) lies on \(AC\) such that \(BD\) (not shown) is perpendicular to \(AC\). What is the value of \(\frac{BC}{BD}\)?
原库参考解析
直线・角与多边形进阶
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・角与多边形进阶
In the figure, two lines intersect at a point. If w = 128, what is the value of z?
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 (9x - 110)°. 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・角与多边形挑战
In triangle ABC, the measure of angle A is 31°, 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・角与多边形挑战
A polygon has exactly 95 sides. If the measure of each of the 95 interior angles of this polygon is (180p)°, what is the value of p?
原库参考解析
直线・角与多边形进阶
In the figure, line l1 is parallel to line l2 and line m1 is parallel to line m2. What is the value of x ?
原库参考解析
直线・角与多边形挑战
The lengths of the two sides of a triangle are 12 inches and 7 inches, respectively. If the length of the third side of the triangle is x inches such that x is an integer, how many possible values can x have?
原库参考解析
