Key Criteria for Triangle Similarity

In 9th grade Geometry, it’s important to know how to tell if two triangles are similar.

Triangle similarity means the triangles have the same shape, but they can be different sizes.

Here are three main ways to prove triangles are similar:

1. Angle-Angle (AA) Similarity

• If two angles in one triangle are the same as two angles in another triangle, then the triangles are similar.
• For example, if $\angle A = \angle D$ and $\angle B = \angle E$, then we say $\triangle ABC$ is similar to $\triangle DEF$.

2. Side-Side-Side (SSS) Similarity

• If the sides of two triangles match up in a certain way, then they are similar.
• This can be shown like this:
  • If $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$, then $\triangle ABC$ is similar to $\triangle DEF$.
• For example, if $AB = 4$, $DE = 2$, $BC = 6$, $EF = 3$, and $CA = 8$, $FD = 4$, all these sides are in proportion.

3. Side-Angle-Side (SAS) Similarity

• If two sides of one triangle are in the right proportion to two sides of another triangle, and the angle between those sides is the same, the triangles are similar.
• We can write this like this:
  • If $\frac{AB}{DE} = \frac{AC}{DF}$ and $\angle A = \angle D$, then $\triangle ABC$ is similar to $\triangle DEF$.

Summary of Triangle Similarity Criteria

• AA: If two angles are equal → Triangles are similar.
• SSS: If the sides match up in proportion → Triangles are similar.
• SAS: If two sides are proportional and the angle between them is equal → Triangles are similar.

Knowing these criteria not only helps with solving your geometry problems, but also strengthens your understanding of shapes in math. It’s important to get good at these ideas if you want to study geometry and other related subjects in the future.