Vertical angles

Vertical angles are a pair of non-adjacent angles formed when two lines or line segments intersect.

In the diagram above, ∠1 and ∠3 are a pair of vertical angles. ∠2 and ∠4 are also a pair of vertical angles. Vertical angles are congruent, therefore ∠1≅∠3 and ∠2≅∠4.

The term "vertical": in this context does not refer to its more well-known meaning referencing an upright position. You can think of vertical angles as a pair of angles oriented in such a way that reflecting one angle across its vertex will line it up with the other angle. Depending on the orientation of the vertical angles, it can look like the letter "X"

Vertical angles are congruent

One way to show vertical angles are congruent is to use pairs of adjacent angles. In the diagram above, since angles 1 and 2 are adjacent and form a straight angle, ∠1 + ∠2 =180°. Also, since angles 2 and 3 are adjacent and form a linear pair then,

∠1 + ∠2 = 180° = ∠2 + ∠3

∠1 = ∠3

So, ∠1≅∠3. Similarly, it can be shown that ∠2≅∠4.

Intersecting chord theorem of angles

When two chords of a circle intersect inside the circle, two pairs of vertical angles are formed. The measure of the angle formed is half the sum of the arcs subtended by the vertical angles formed by the chords of the circle:

Referencing the diagram above, . This is true for any vertical angles formed by two chords inside the circle. The chords do not have to intersect at the center of the circle for this theorem to be true.