这部分需要会什么？
- 根据对应角或对应边判定相似
- 按顶点顺序写出正确的边长比例
- 区分相似比例与全等关系
核心概念与公式
相似三角形对应角相等、对应边成比例。常用判定包括两角相等（AA）、两组对应边成比例且夹角相等（SAS）、三组对应边成比例（SSS）。全等图形形状和大小都相同，对应边比为 1。
若 \(\triangle ABC\sim\triangle DEF\)，顺序表示 \(A\leftrightarrow D\)、\(B\leftrightarrow E\)、\(C\leftrightarrow F\)。长度倍率为 \(k\) 时，周长倍率为 \(k\)，面积倍率为 \(k^2\)。
做题时怎样使用
- 先从角标记或题目给出的顶点顺序找对应关系
- 用同一方向写各组边长比，并选已知量最多的一组
- 求面积时将长度倍率平方；需要全等时进一步检查倍率是否为 1
容易混淆的地方
两条边的比值相同，并不足以单独证明三角形相似。不要按图上“左边、右边”机械配对，图形可能旋转或翻转。
用一个例题串起方法
知识点教学例题两个相似三角形的对应边长为 6 和 9。较小三角形的面积为 20，求较大三角形的面积。
展开解题思路 ＋
从小到大的长度倍率为 \(9/6=3/2\)，面积倍率为 \((3/2)^2=9/4\)。较大面积为 \(20\times9/4=45\)。
本知识点题库
先独立作答，再查看答案与解析。
相似・全等与比例线段挑战
The table gives the areas and perimeters of two similar rectangles, where n is a constant.
What is the value of n?
原库参考解析
相似・全等与比例线段进阶
Triangle TUV is dilated by a scale factor of \( \frac{1}{2} \) to obtain triangle T'U'V' (not shown), where T corresponds to T' and U corresponds to U'. What is the measure, in degrees, of angle V'?
原库参考解析
相似・全等与比例线段进阶
Triangle TUV is dilated by a scale factor of \( \frac{1}{2} \) to obtain triangle T'U'V' (not shown), where T corresponds to T' and U corresponds to U'. What is the measure, in degrees, of angle V'?
原库参考解析
相似・全等与比例线段挑战
In triangle ABC and triangle XYZ , the measures of angles A and X are each \( 20^{∘} \) , and the lengths of \( AB \) and \( XY \) are each 16 centimeters. Which of the following additional pieces of information is(are) sufficient to prove that triangle ABC is congruent to triangle XYZ?
I. The measures of angles C and Z are each \( 25^{∘} \).
II. \( \frac{\overline{AB}}{\overline{AC}}=\frac{\overline{XY}}{\overline{XZ}} \)
原库参考解析
相似・全等与比例线段挑战
In triangle ABC, the measure of angle A is 56° and AC = 30. In triangle PQR, the measure of angle P is 56° and PR = 90. Which additional piece of information is sufficient to prove that triangle ABC is similar to triangle PQR?
原库参考解析
相似・全等与比例线段挑战
Triangle ABC is similar to triangle XYZ, where A and B correspond to X and Y, respectively. The length of each side of triangle XYZ is k times the length of its corresponding side in triangle ABC, where k is an integer greater than 1. The measure, in degrees, of angle Z can be represented by the expression \( 5x+28 \), where x is an integer. Which of the following expressions represents the measure, in degrees, of angle C for all possible values of x?
原库参考解析
相似・全等与比例线段进阶
In the figure, line m is parallel to line n, and lines AE and CD intersect at point B. Which additional piece of information is sufficient to prove that triangle ABC is congruent to triangle EBD?
原库参考解析
相似・全等与比例线段进阶
Triangle ABC is similar to triangle XYZ, such that A, B, and C correspond to X, Y, and Z, respectively. The length of each side of triangle XYZ is 2 times the length of its corresponding side in triangle ABC. The measure of angle B is 17°. What is the measure of angle Y, in degrees?
原库参考解析
相似・全等与比例线段挑战
The table gives the areas and perimeters of two similar rectangles, where n is a constant.
What is the value of n?
参考答案
相似・全等与比例线段挑战
In isosceles triangle \( ABC \) sides \( AB \) and \( AC \) are congruent. Point D divides side \( BC \) such that the length of \( BD \) is \( \frac{2}{3} \) of the length of \( BC \). Point E lies on side \( AB \) and point F lies on side \( AC \) such that when segments \( DE \) and \( DF \) are drawn, angle \( BED \) is congruent to angle \( CFD \). If the length of \( BE \) is 12, what is the length of \( CF \)?
参考答案
相似・全等与比例线段挑战
Each side of equilateral triangle R is multiplied by a scale factor of k to create equilateral triangle U. The length of each side of triangle U is \( \frac{1}{31} \) greater than the length of each side of triangle R. Which of the following could be the value of k?
参考答案
相似・全等与比例线段进阶
Trapezoid CDEF is similar to trapezoid JKLM, such that C, D, E, and F correspond to J, K, L, and M, respectively. The length of each side of trapezoid JKLM is 4 times the length of its corresponding side in trapezoid CDEF. The measure of angle C is 42°. What is the measure of angle J?
参考答案
