Theorems: Angles between Two Lines

Adjacent means “next to.” But we use this word in a very specific way when we refer to adjacent angles. Study these two figures. Only the pair on the right is considered to be adjacent, angles c and d. Adjacent angles must share a common side and a common vertex, and they must not overlap each other.

Vertical angles are pairs of angles formed by two intersecting lines. Vertical angles are not adjacent angles — they are opposite each other. In this diagram, angles a and c are vertical angles, and angles b and d are vertical angles. Vertical angles are congruent.

These two lines are parallel, and are cut by a transversal, which is just a name given to a line that intersects two or more lines at different points. Eight angles appear, in four corresponding pairs that have the same measure, so therefore are congruent.

These four corresponding pairs are:

angles a and e
angles c and g
angles b and f
angles d and h

The angles that lie in the interior area, or the area between the two lines that are cut by the transversal, are called interior angles. Angles c, d, e and f are interior angles. Angles a, b, g, and h lie in the exterior area, and they are called “exterior angles.”

We call angles on opposite sides of the transversal alternate angles. Angles c and f, and d and e, are alternate interior angles. Angles a and h, and b and g, are alternate exterior angles. Note that these alternate pairs are also congruent.

When a transversal cuts two lines that are not parallel, as shown here, it still forms eight angles—four corresponding pairs. However, the corresponding pairs are not congruent as occurs with parallel lines.

What is Vertically Opposite Angles?

When two straight lines intersect each other four angles are formed.

The pair of angles which lie on the opposite sides of the point of intersection are called vertically opposite angles.

In the given figure, two straight lines AB and CD intersect each other at point O. Angles AOD and BOC form one pair of vertically opposite angles; whereas angles AOC and BOD form another pair of vertically opposite angles.

Vertically opposite angles are always equal.
i.e., ∠AOD = ∠BOC
and ∠AOC = ∠BOD

Note:
In the given figure; rays OM and ON meet at O to form ∠MON (i.e. ∠a) and reflex ∠MON (i.e. ∠b). It must be noted that ∠MON means the smaller angle only unless it is mentioned to take otherwise.

For example,in the given figure, two lines GH and KL are intersecting at a point P.

We observe that with the intersection of these lines, four angles have been formed. Angles ∠1 and ∠3 form a pair of vertically opposite angles; while angles ∠2 and ∠4 form another pair of vertically opposite angles.
Clearly, angles ∠1 and ∠2 form a linear pair.
Therefore, ∠1 + ∠2 = 180°
or, ∠1 = 180° – ∠2 …………(i)

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