Congruence: conditions for congruent triangles

​​In a nutshell

Congruent shapes are shapes that are exactly the same apart from rotational/mirror differences. There are four main conditions for two triangles to be congruent.

Congruency

Two shapes are congruent if they have the same side lengths and the same angles. It doesn't matter if the shapes are rotated slightly differently or if they're mirrored.

Congruency conditions

There are four ways to prove that two triangles are congruent.

name	description	example
SSS	All three sides of the triangle are the same
SAS	Two sides and the angle between them are the same
AS	Two angles and a corresponding side are the same
RHS	Both triangles have a right-angle, the same hypotenuse and other common side

Example 1

Are the two triangles shown congruent to one another?

At first glance, the two triangles look to be congruent due to the ASA rule - both triangles have angles of $$40^\circ$$ and $$80^\circ$$ and they both have a side of $$2.2cm$$.

However, upon closer inspection, this is not a case of ASA. This is because the two sides aren't corresponding.

The triangle to the left has a side length of $$2.2cm$$ that is between the two angles of $$40^\circ$$ and $$80^\circ$$.

The triangle to the right has a side length of $$2.2cm$$ that is not between the two angles of $$40^\circ$$ and $$80^\circ$$, so it's not the corresponding sides that are equal.

The triangles are NOT congruent.