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


1. Prove that the two triangles are congruent.


Show Answer

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.


Show Answer

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.

Hence, we can solve x algebraically:

2x+17^\circ=53^\circ \\ 2x=36^\circ \\ x=18^\circ