Explain angle pairs ?

Adjacent angle pairs
Two angles are adjacent if they:
1. Have the same vertex.
2. Share a common side.
3. Have no interior points in common.

Definition 2 Vertical angle pairs (opposite angle pairs)
Vertical angles are a pair of nonadjacent angles formed by two intersecting lines. The vertical angle pairs are also called opposite angle pairs.

Example 1 :  Name all pairs of vertical angles.

Solution: vertical angle pairs: 1 and 4; 2 and 5; 3 and 6.

Definition 3 Complementary angles
Two angles are complementary if the sum of their measures is 90°. If °A is complementary to B, then mA + mB = 90, and each angle is called the complement of the other angle.
Definition 4 Supplementary angles
Two angles are supplementary if the sum of their measures is 180°. If A is supplementary to B, then mA + mB = 180°, and each angle is called the supplement of the other angle.
*** A trick for remembering complementary and supplementary angles: Notice that s is greater than c in alphabetical order. Therefore, s is the greater angle, 180°; c is the smaller angle, 90°.

Definition 5 Linear pair

Two angles are a linear pair if they have a common side and their other sides are opposite rays.
Postulate 3.1 (Linear Pair Postulate)
If two angles are a linear pair, then they are supplementary.

Example 2

Determine the measure of an angle if it exceeds twice the measure of its complement by 60.
Solution: Let x = measure of angle
then 90 - x = measure of the complement of the angle
x = 2(90 - x) + 60
x = 180 - 2x + 60
x = 240 - 2x
3x = 240
x = 80
Therefore, measure of angle = 80°

Example 3

A student wonders whether the following two assertions are true:
(a). If a pair of angles are complementary, then they must be adjacent.
(b). If the exterior sides of a pair of adjacent angles form a straight line, then the angles are supplementary.

Please tell whether they are true or false. You may draw diagrams to prove your belief. Solution:
(a) A pair of complementary angles may be either adjacent or nonadjacent as shown in the following figure. The assertion is false.
(b) The assertion is true since a straight line is formed which implies that the sum of the measures of the adjacent angles is 180°. Therefore, the angles are supplementary.