Students will learn what the third angle theorem is, why it's true, and how to use it. Your students must know the angles of a triangle add to 180°. Watch this or this to get an idea of how to introduce this theorem. After that, you can show students this proof. To understand the proof, students must know the angles in a triangle sum to \(180^\circ.\) The heart of the proof is realizing that if \(a + b + x = 180^\circ,\) and \(a + b + y = 180^\circ,\) then we can put those two equations together, yielding:

$$a + b + x = a + b + y$$Removing \(a + b\) from both sides, we find \(x = y.\) Finally, here are some example problems:

## Arithmetic

Determine if the triangles below are similar (source).

## Algebraic

Solve for \(x\) (source).

## Geometric

Here are two geometric problems. Students must know the vertical angles theorem and all the theorems involving parallel lines being intersected by a transversal.

Prove \(\triangle RSV \sim \triangle TUV\) (source).

Note: Khan Academy doesn't cover this topic.