Similar Triangles

Similar triangles rules

Two triangles are considered to be similar if they follow these two rules:

• They have the same size corresponding angles.
• All corresponding side lengths have the same ratio.

Similar triangles proof

The idea of similar triangles can be shown and explained in the following diagram:

Example of similar triangles, StudySmarter Originals

Above you can see that the two triangles have a corresponding angle. As well both triangles have sides that are the same ratio. This means that the triangle side lengths are in proportion with one another, the bigger triangle on the right, is 2 times bigger than the smaller triangle on the left. This ratio is also known as a scale factor.

There are different theorems that can further prove the idea of similar triangles:

• SSS similarity theorem

• AA similarity theorem

• SAS similarity theorem

SSS similarity theorem

The SSS similarity theorem suggests that when three sides of one triangle are proportional to a corresponding triangle, the triangle is similar.

This theorem can be represented in the following formula:

$\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}$

AA similarity theorem

The AA similarity theorem suggests that when the two angles in one triangle are equal to the two angles in another triangle, both triangles are similar.

$\angle A=\angle X,\angle B=\angle Y,\angle C=\angle Z$

SAS similarity theorem

The SAS similarity theorem suggests that when the included angle of one triangle is equal to the included angle of another triangle and the sides length of both triangles are proportional, the triangle will be similar.

Similar triangles formulas

When looking at similar triangles, they are often made clear to us by using the$~$symbol. Formulae can be used to show each of the similar triangle theorems:

• When$\angle A=\angle X,\angle B=\angle Y,\angle C=\angle Z$, $△ABC~△XYZ$
• When $\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}$ , $△ABC~△XYZ$
• When $\frac{AB}{XY}=\frac{BC}{YZ}\mathrm{and}\angle B=\angle Y$, $△ABC~△XYZ$

Types of similar triangle examples