Geometry

UK Secondary (7-11)

Proofs Using Congruent Triangles

Lesson

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.

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

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

`B``A``D`=∠`B``C``D`.In $\triangle BAD$△

`B``A``D`and $\triangle BCD$△`B``C``D`we have:

$ABCD$`A``B``C``D` is a parallelogram with $AE=FC$`A``E`=`F``C`. Prove that $DE$`D``E`=$FB$`F``B`.

In $\triangle AED$△

`A``E``D`and $\triangle CFB$△`C``F``B`we have :