# Angle Relationships

Many angle relationships involve exactly a pair of angles. These relationships can be classified under the following categories:

**1.** Relationships based on the Measurement of Angles**2.** Relationships based on the Position of Angles

### Relationships based on the measurement of angles

On the basis of the measurement of the angles, the angles can be categorized into the following two categories:

Complementary Angles: Those angles for which the sum of the measures is $90^o$. That is, if $\angle$A + $\angle$B = $90^o$, each angle is the complement of the other.

Supplementary Angles: Those angles for which the sum of the measures is $180^o$. That is, if $\angle$A + $\angle$B = $180^o$, then each angle is the supplement of the other.

Congruent Angles: A pair of angles are said to be congruent angles if they have the same measure.

Notation: If two angles $\angle$A and $\angle$B are congruent, we denote it by $\angle$A $\cong$ $\angle$B. It is customary to denote angles by small letters of the Greek Alphabet. That is, we use the Greek letters $\alpha$, $\beta$, $\gamma$ and so on to denote angles. Thus, if $\alpha$ and $\beta$ are congruent angles, we write $$\alpha \cong \beta$$

### Relationships based on the Position of Angles

On the basis of the position of the angles, every pair of angles can be categorized into the following categories:

Adjacent Angles: Adjacent angles are a pair of angles which share a common vertex and side, but have no common interior points..

Complementary angles and supplementary angles can be adjacent angles or non-adjacent angles.

Vertical Angles: Vertical angles are two non-adjacent angles formed by two intersecting lines.

In the figure, $\angle$1 and $\angle$2 are a pair of vertical angles. Another pair of vertical angles is $\angle$3 and $\angle$4.

Linear Pair of Angles: A linear pair is a pair of adjacent angles whose non-common sides are opposite rays.

In the figure,

$\angle$1 and $\angle$2 are vertical angles.

$\angle$3 and $\angle$4 are vertical angles.

$\angle$1 and $\angle$3 is a linear pair of angles.

$\angle$3 and $\angle$2 is a linear pair of angles.

$\angle$2 and $\angle$4 is a linear pair of angles.

$\angle$4 and $\angle$1 is a linear pair of angles.

Vertical angles are always congruent.

Thus, we have $\angle$1 $\cong$ $\angle$2 and $\angle$3 $\cong$ $\angle$4

Linear Pair Postulate: If two angles form a linear pair, then the angles are supplementary.

From the above diagram, it is clear that

m $\angle$1 + m $\angle$3 = $180^o$, m $\angle$2 + m$\angle$3 = $180^o$, m $\angle$2 + m $\angle$4 = $180^o$, and m $\angle$ 1 + m $\angle$4 = $180^o$