# Congruency and Similarity

## Congruent Triangles

Two triangles are congruent if they are the same shape and size. All pairs of corresponding sides and angles are equal.

For example:

When the two triangles are congruent, the congruent notation is:

$\triangle ABC\equiv \triangle DEF$

## Congruence Test

There are 4 tests to check whether two triangles are congruent or not.

(courtesy of learnhive.net)

### Examples:

1. Prove that the two triangles are congruent.

Out of the 4 congruence tests, the best to use is ASA, although the side is not an included side. Therefore:

$\angle C=\angle D\qquad (angle)\\ \angle A=\angle E\qquad (angle)\\ AB=EF\qquad (side)\\ \\ \therefore \triangle ABC \equiv \triangle DEF$

2. Find the value of x given that $\triangle ABC \equiv \triangle DEF$.

First of all, we know that $AB=PR$ and $BC=RQ$.
Therefore, $\angle B=\angle R$.
Now, we know that $\angle R = 180^\circ-60^\circ-67^\circ = 53^\circ$.
$2x+17^\circ=53^\circ \\ 2x=36^\circ \\ x=18^\circ$