Included are the axioms and theorems used to prove that two vertical angles are congruent.

Axioms of Angle Measure

  1. Right angle measures \(90^\circ\)

  2. \(m\angle ABC=m\angle CBA\)

  3. If D is the interior of \(\angle ABC\), then \(m\angle ABC=m\angle ABD+m\angle BDC\)

  4. There exists a unique ray that is the angle bisector of \(\angle ABC\).

Congruence and Angle Measure Theorem

\(\angle ABC\cong \angle DEF \Leftrightarrow m\angle ABC=m\angle DEF\)

  1. If two angles are congruent then the measure of those two angles is the same.

  2. If the measure of two angles is the same, then those two angles are congruent.

(Note that the measure for angles that will be used on the proof will be in degrees. )

Supplementray Interior Angle Theorem

If two lines are parallel then the supplementary interior angles add to \(180^\circ\).

**Fig. 4: ** Supplementary interior angles are shown in red.

Fig. 4: Supplementary interior angles are shown in red.

In Fig. 4 the supplementary interior angles \(\angle AEF\) and \(\angle CFE\) are shown in red and add up to \(180^\circ\).