# Proofs Using Similar Triangles

Lesson

Recall that if two triangles are similar, then one is an enlargement or reduction of the other.

Similar triangles have two very important features:

• All corresponding angles are equal.

• All pairs of corresponding sides are in the same ratio.

$\frac{A}{D}$AD$=$=$\frac{B}{E}$BE$=$=$\frac{C}{F}$CF

There are three main methods we can use to prove that two triangles are similar.

• AAA (Angle, Angle, Angle) - All corresponding angles are equal.

• SAS (Side, Angle, Side) - Two pairs of corresponding sides are in the same ratio and the corresponding angles in between are equal.

• SSS (Side, Side, Side) - All three pairs of corresponding sides are in the same ratio.

$\frac{A}{D}$AD$=$=$\frac{B}{E}$BE$=$=$\frac{C}{F}$CF

You can revise these ideas here.

## Similar triangles in more general proofs

We're going to use our knowledge of similar triangles along with other geometric concepts to come up with some more general proofs.

#### Worked Examples

##### Question 1

Consider the following parallelogram.

a) Prove that $\triangle ABO\simeq\triangle DOC$ABODOC.

b) Hence, deduce that the diagonals of a parallelogram bisect each other.

##### Solution

a) We are going to prove that $\triangle ABO$ABO and $\triangle DOC$DOC are similar using an AAA proof.

Firstly, we know that $\angle AOB=\angle DOC$AOB=DOC because they are vertically opposite.

Secondly, we know that $AB\parallel DC$ABDC, because $ABCD$ABCD is a parallelogram. Hence, $\angle OAB=\angle OCD$OAB=OCD because they are alternate angles.

Thirdly, $\angle ABO=\angle ODC$ABO=ODC for the same reason. They are alternate angles on parallel lines.

Hence, by our AAA proof, $\triangle ABO\simeq\triangle DOC$ABODOC.

b) We know that pairs of corresponding sides of similar triangles are all in the same ratio.

$\frac{AB}{DC}$ABDC$=$=$\frac{BO}{DO}$BODO$=$=$\frac{AO}{CO}$AOCO

But we also know that $ABCD$ABCD is a parallelogram, and parallelograms have equal opposite sides. Hence, $AB=DC$AB=DC, which means that $\frac{AB}{DC}=1$ABDC=1. Therefore, all pairs of corresponding sides are in the same ratio $1$1, in other words, equal.

So these triangles aren't just similar. They are congruent.

So, as we can see, the diagonals split each other into two equal halves. $AO=CO$AO=CO and $DO=BO$DO=BO. Hence, the diagonals of a parallelogram bisect each other.