In Figure 1 two intersecting straight lines are shown. They form four angles ,, and .
Pair of angles and are vertical. Pair of angles and are vertical as well.
Two angles are called vertical if the sides of one of them are continuations of sides of the other.
Theorem (Vertical angles theorem)
Vertical angles are congruent.
Referring to Figure 1, we are going to prove that = and =.
+ = 180° (because angles and make in sum the straight angle), and
+ = 180° (because angles and make in sum the straight angle).
+ = 180° (because and make in sum the straight angle) and
+ = 180° (because and make in sum the straight angle).
The proof is completed.
Figure 1. Vertical angles
If in Figure 1 one of vertical angles = 37°,
find three other angles , and .
= 37° as the vertical angle to ;
= 180° -37° = 143° as the complementary angle to ;
= 143° as the vertical angle to .