Congruence and Similarity

Ontario 10 Academic (MPM2D)

Similarity in triangles

Lesson

As the name suggests, similar triangles share common features but are not exactly the same. In other words, a triangle will either be enlarged or shrunk but the angles will always be the same.

In similar triangles:

- all corresponding angles are equal
- all corresponding sides are in the same ratio

Just like congruent triangles, similar triangles may be transformed, so take your time working out whether triangles are similar.

Careful!

Congruent triangles are always similar but similar triangles are not always congruent.

There are three distinct methods for proving two triangles are similar. You only need to use one of these proofs to show two triangles are similar, then all the features of similar triangles can be applied.

This proof is also sometimes called *equiangular*, which means all corresponding angles in similar triangles are equal. However, we only have to prove two pairs of corresponding angles are equal because the third angle must be equal since the angle sum of a triangle is always $180^\circ$180°.

The triangles below show that all three corresponding pairs of angles are equal. Therefore, we can say that $\triangle ABC$△`A``B``C` is similar to $\triangle PQR$△`P``Q``R` because they are equiangular.

If two triangles have two pairs of sides in the same ratio and equal included angles, then these triangles are similar.

For example, we can prove the triangles below are similar using the SAS proof. There are two pairs of sides in the same ratio ($\frac{PQ}{AB}=\frac{QR}{BC}$`P``Q``A``B`=`Q``R``B``C`$=$=$2$2) and the included angle (marked by the dot) is equal.

If two triangles have all three pairs of corresponding sides in the same ratio, then these triangles are similar.

For example, all corresponding sides in the triangle below are in the same ratio:

$\frac{12}{4}=\frac{12}{4}$124=124$=$=$\frac{21}{7}$217$=$=$3$3

Which of these triangles are similar?

Consider the two similar triangles.

a) $DE$`D``E` corresponds to which side in $\triangle LMN$△`L``M``N`?

b) $DC$`D``C` corresponds to which side in $\triangle LMN$△`L``M``N`?

a) Why is $BE$`B``E` parallel to $CD$`C``D`?

b) Hence form a statement in which you identify the angle that is equal to $\angle BEA$∠`B``E``A`.

c) Why is $\triangle ABE\simeq\triangle ACD$△`A``B``E`⫻△`A``C``D`?

d) State the enlargement factor going from the smaller triangle to the larger triangle.

e) Find the value of the variable.

Verify, through investigation, the properties of similar triangles