Before hopping into our proof, lets look at a few definitions with some Geogebra images.

Vertical Angles

In class we defined vertical angles as being “across from each other”.

**Fig. 1: ** Vertical angles are shown in pink.

Fig. 1: Vertical angles are shown in pink.

We can see a simple example of this in Fig. 1 where a pair of pink angles \(\angle AOB\) and \(\angle COD\) are vertical to each other.

Note that \(\angle AOC\) and \(\angle BOD\) are vertical angles as well.

Congruence

\(x\cong y\) if there is an \(\underline{\text{isometry}}\) that superimposes x onto y.

  • Isometry is a map that preserves distance and angles

    • translation (move without turning)

    • rotation (moving about a fixed point)

    • reflection (mirror)

    • combination

**Fig. 2: ** Two congruent triangles

Fig. 2: Two congruent triangles

In Fig. 2 we see the two triangles are congruent, and would only need a translation isometry or two to map \(\triangle ABC\) onto \(\triangle A_1B_1C_1\).

Supplementary Angles

We defined supplementary angles as angles whose measurement adds up to \(180^\circ\).

**Fig. 3: **Supplementary angles are shown in pink and orange.

Fig. 3: Supplementary angles are shown in pink and orange.

In Fig.3 we can clearly see that \(m\angle AOC\) shown in pink and \(m\angle AOB\) in orange adds to a straight line or \(180^\circ\). We can also see three other pairs of supplementary angles:

  • \(m\angle AOB+m\angle BOD=180^\circ\)

  • \(m\angle BOD+m\angle COD=180^\circ\)

  • \(m\angle COD+m\angle AOC=180^\circ\)