Triangle Similarity Theorems

Last updated on 26 October 2025

Table of contents

AA (Angle-Angle) Similarity Theorem
SSS (Side-Side-Side) Similarity Theorem
SAS (Side-Angle-Side) Similarity Theorem
Solved Examples on Triangle Similarity Theorems

Triangle similarity theorems are essential principles in geometry that help establish relationships between different triangles. They are particularly important in understanding the proportional relationships between corresponding sides and angles of similar triangles. When two triangles are similar, their corresponding angles are equal, and their corresponding sides are in proportion.
There are three main triangle similarity theorems:

AA (Angle-Angle) Similarity Theorem

This theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. It implies that the corresponding sides are proportional, but it does not determine the scale factor. This means that the two triangles have the same shape but possibly different sizes.

For example, consider above two triangles, △ABC and △PQR.

∠ B

is congruent to

∠ Q

and

∠ C

is congruent to

∠ R

, hence according to the AA Similarity Theorem, these two triangles are similar. This similarity implies that the remaining angles,

∠ A

and

∠ P

, are also congruent, although their corresponding sides may not be equal.

SSS (Side-Side-Side) Similarity Theorem

This theorem states that if the corresponding sides of two triangles are proportional, then the triangles are similar. This theorem implies that all three pairs of corresponding sides are in proportion.

For example, consider above two triangles, △ABC and △PQR. If the ratio of the lengths of the corresponding sides AB to PQ, BC to QR, and AC to PR are equal, i.e. if

\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}

, then according to the SSS Similarity Theorem, these two triangles are similar.

SAS (Side-Angle-Side) Similarity Theorem

This theorem states that if two of the sides of two triangles are proportional and the angles between those sides are congruent, then the triangles are similar.

For example, consider above two triangles, △ABC and △PQR.

∠ B

is congruent to

∠ Q

, i.e.

∠ B = ∠ Q

, and if the length of side AB is proportional to the length of side PQ, and the length of side BC is proportional to the length of side QR, i.e. if

\frac{AB}{PQ} = \frac{BC}{QR}

, then according to the SAS Similarity Theorem, these two triangles are similar.

These theorems play a crucial role in various geometric and real-world applications. They provide the foundation for solving problems related to indirect measurement, map scaling, and proportionality. Understanding triangle similarity theorems is essential in fields such as engineering, architecture, and physics, where scaling and proportionality are integral to accurate design and analysis.

Solved Examples on Triangle Similarity Theorems

Example 1: Prove that the right-angled triangles, △PQR and △LMN in the figure below are similar.

Solution:

Given:

In the figure above $△ PQR$ and $△ LMN$ are right-angled triangles, right angled at $∠ Q$ and $∠ M$ respectively.

∴ ∠ Q = ∠ M = 90 °

Also, as shown in the figure, $∠ R$ and $∠ N$ measure $x °$ .

∴ ∠ R = ∠ N = x °

Two pairs of angles i.e. $∠ Q$ , $∠ M$ and $∠ R$ , $∠ N$ are congruent in $△ PQR$ and $△ LMN$ , Therefore by AA (Angle-Angle) similarity theorem these triangles are similar.

∴ △ PQR \sim △ LMN

Example 2: Prove that the triangles, △XYZ and △RST in the figure below are similar.

Given: As shown in the figure above, in

△ XYZ

XY = 24 units, YZ = 10 units, XZ = 26 units

and in

△ RST

RS = 12 units, ST = 5 units, RT = 13 units

Let's check the ratio of the side lenghts of the triangles given.

$\begin{array}{r} \frac{XY}{RS} = \frac{24}{12} = \frac{2}{1} \\ \frac{YZ}{ST} = \frac{10}{5} = \frac{2}{1} \\ \frac{XZ}{RT} = \frac{26}{13} = \frac{2}{1} \end{array}$

All the corresponding sides of the given triangle are in the ratio of 2:1.

$∴ \frac{XY}{RS} = \frac{YZ}{ST} = \frac{XZ}{RT}$

Therefore, by SSS (Side-Side-Side) similarity theorem, the given triangles are similar.

∴ △ XYZ \sim △ RST

Example 3: Prove that the triangles, △ABC and △PQR in the figure below are similar.

Given: As shown in the figure above, in

△ ABC

\begin{aligned} AB & = & 6 units \\ BC & = & 12 units \\ ∠ B & = & x ° \end{aligned}

and in

△ PQR

\begin{aligned} PQ & = & 4 units \\ QR & = & 8 units \\ ∠ Q & = & x ° \end{aligned}

In above triangles,

∠ B

and

∠ Q

are equal.

$∠ B = ∠ Q = x °$

Let's check the ratio of the sides adjacent to

∠ B

and

∠ Q

\begin{matrix} \frac{AB}{PQ} = \frac{6}{4} = \frac{3}{2} \\ \frac{BC}{QR} = \frac{12}{8} = \frac{3}{2} \end{matrix}

Since $\frac{AB}{PQ} = \frac{BC}{QR}$ and $∠ B = ∠ Q$ , by SAS (Side-Angle-Side) similarity theorem, the given triangles are similar.

∴ △ ABC \sim △ PQR

Triangle Similarity Theorems

AA (Angle-Angle) Similarity Theorem

SSS (Side-Side-Side) Similarity Theorem

SAS (Side-Angle-Side) Similarity Theorem

Solved Examples on Triangle Similarity Theorems

Triangle Inequality Theorem

Alternate Angles