Congruent triangles are triangles that are identical in shape and size. In other words, all corresponding sides and angles are equal. You can prove two triangles are congruent using one of four proofs.

The Four Proofs of Congruency

Here are the four proofs of congruency. You only need to use one of the to prove two triangles are congruent.

SSS- all three pairs of corresponding sides are equal

AAS- two pairs of corresponding angles and one pair of corresponding sides are equal

SAS- two pairs of corresponding sides, and the pair of included angles are equal

RHS- right-angled triangles with equal hypotenuses and a pair of equal corresponding sides

Sometimes we are given information that indicates sides are the same lengths or that angles are the same. However, sometimes we're not given this information. So we need to use our geometrical knowledge, such as the properties of shapes to help us work out which sides and angles are corresponding so we can prove two triangles are congruent.

In the first video example below, we are given information on the diagram indicating pairs of equal sides.

In the second video example, however, we need to use our knowledge of properties of shapes to identify angles and sides that are equal.

Worked Examples

Question 1

In the diagram, $AB=CB$AB=CB and $D$D is the midpoint of side $AC$AC.

Without using the properties of an isosceles triangle show that $\angle BAD=\angle BCD$∠BAD=∠BCD.

In $\triangle BAD$△BAD and $\triangle BCD$△BCD we have:

Question 2

$ABCD$ABCD is a parallelogram with $AE=FC$AE=FC. Prove that $DE$DE=$FB$FB.

In $\triangle AED$△AED and $\triangle CFB$△CFB we have :

Outcomes

9.G.T.1

Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence). Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence). Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).

9.G.T.2

Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle. The angles opposite to equal sides of a triangle are equal. The angles opposite to equal sides of a triangle are equal. Triangle inequalities and relation between ‘angle and facing side’; inequalities in a triangle.

9.G.Q.1

The diagonal divides a parallelogram into two congruent triangles. In a parallelogram opposite sides are equal and conversely. In a parallelogram opposite angles are equal and conversely. A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal. In a parallelogram, the diagonals bisect each other and conversely.

Proofs Using Congruent Triangles