First, students will learn what the third angle theorem is.

The Third Angle Theorem: Lesson (Geometry Concepts) by CK-12 Foundation

Next, students will learn why the third angle theorem is true. 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 \(x = y.\)

Congruent Polygons & Third Angle Theorem by ProfRobBob

After that, students will learn how to use the third angle theorem. 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).

Conclude by giving your students these challenges:

- Unit Fractions by NRICH
- Swimming Pool by NRICH
- Coins on a Plate by NRICH
- Counting Factors by NRICH