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
In the figure shown, triangle \( CAE \) is similar to triangle \( CBD \). The measure of angle \( CBD \) is 56°, and \( AE=24\left(BD\right) \). What is the measure of angle \( CAE \)?
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 52° and AC = 35. In triangle PQR, the measure of angle P is 52° and PR = 105. 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・全等与比例线段进阶
In the figure shown, triangle CAE is similar to triangle CBD. The measure of angle CBD is 56°, and AE = 22(BD). What is the measure of angle CAE?
原库参考解析
相似・全等与比例线段挑战
In the figure shown, \( WZ \) and \( XY \) intersect at a point \( Q \). \( YQ=27 \), \( WQ=90 \), \( WX=60 \), and \( XQ=110 \). What is the length of \( YZ \) ?
原库参考解析
相似・全等与比例线段进阶
In the figure shown, triangle CAE is similar to triangle CBD. The measure of angle CBD is \( 57° \), and \( AE=21\left(BD\right) \).What is the measure of angle CAE?
原库参考解析
相似・全等与比例线段挑战
In triangle \( ABC \), point \( D \) is the midpoint of segment \( AB \) and point \( E \) is the midpoint of segment \( BC \). If \( BD=8 \) and \( DE=14 \). What is the length of segment \( AC \) ?
原库参考解析
相似・全等与比例线段挑战
Parallelogram \( PQRS \) is similar to parallelogram \( TUVW \), such that \( P \), \( Q \), \( R \), and \( S \) correspond to \( T \), \( U \), \( V \), and \( W \), respectively. If \( PQ=28TU \), then which of the following must be true?
原库参考解析
相似・全等与比例线段挑战
Rectangles ABCD and EFGH are similar. The length of each side of EFGH is 10 times the length of the corresponding side of ABCD. The area of ABCD is 140 square units. What is the area, in square units, of EFGH?
原库参考解析
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 \( \frac{2}{7} \) 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 14, what is the length of \( CF \)?
原库参考解析
相似・全等与比例线段进阶
In triangle DEF, the measure of angle D is 47° and the measure of angle E is 97°. In triangle RST, the measure of angle R is 47° and the measure of angle S is 97°. Which of the following additional pieces of information is needed to determine whether triangle DEF is similar to triangle RST?
原库参考解析
相似・全等与比例线段进阶
Each side of square A has a length of 13 inches. Each side of square A is multiplied by a scale factor of 3 to create square B. What is the length, in inches, of each side of square B?
原库参考解析
相似・全等与比例线段挑战
In triangle ABC and triangle DEF , sides AB and DE each have a side length of 10 inches, and angles A and D each have an angle measure of 40°. Which of the following additional pieces of information is(are) sufficient to prove whether triangle ABC is congruent to triangle DEF?
I. The measures of angles B and C are equal.
II. The lengths of sides AC and DF are equal.
III. The lengths of sides BC and EF are equal.
原库参考解析
