Certain angle pairs are given special names based on their relative position to one another or based on the sum of their respective measures.

**Adjacent angles**

**Adjacent angles** are any two angles that share a common side separating the two angles and that share a common vertex. In Figure 1

**Figure 1 **Adjacent angles.

## Vertical angles

**Vertical angles** are formed when two lines intersect and form four angles. Any two of these angles that are *not* adjacent angles are called vertical angles. In Figure 2*l* and line *m* intersect at point *Q*, forming ∠1, ∠2, ∠3, and ∠4.

**Figure 2**** **

- Vertical angles:

- ∠1 and ∠3
- ∠2 and ∠4

- Adjacent angles:

- ∠1 and ∠2
- ∠2 and ∠3
- ∠3 and ∠4
- ∠4 and ∠1

*Theorem 7:* Vertical angles are equal in measure.

## Complementary angles

**Complementary angles** are any two angles whose sum is 90°. In Figure 3*ABC* is a right angle, *m* ∠1 + *m* ∠2 = 90°, so ∠1 and ∠2 are complementary.

**Figure 3**Adjacent complementary angles.

Complementary angles do not need to be adjacent. In Figure 4*m* ∠3 + *m* ∠4 = 90°, ∠3, and ∠4, are complementary.

**Figure 4**Nonadjacent complementary angles

**Example 1:** If ∠5 and ∠6 are complementary, and *m* ∠5 = 15°, find *m* ∠6.

Because ∠5 and ∠6 are complementary,

*Theorem 8:* If two angles are complementary to the same angle, or to equal angles, then they are equal to each other.

Refer to Figures 5 *A* and ∠ *B* are complementary. Also, ∠ *C* and ∠ *B* are complementary. *Theorem 8* tells you that *m* ∠ *A* = *m* ∠ *C*. In Figure 6*A* and ∠ *B* are complementary. Also, ∠ *C* and ∠ *D* are complementary, and *m* ∠ *B* = *m* ∠ *D. Theorem 8* now tells you that *m* ∠ *A* = *m* ∠ *C*.

**Figure 5**Two angles complementary to the same angle

**Figure 6 **Two angles complementary to equal angles

**Supplementary angles** are two angles whose sum is 180°. In Figure *ABC* is a straight angle. Therefore *m* ∠6 + *m* ∠7 = 180°, so ∠6 and ∠7 are supplementary.

**Figure 7 **Adjacent supplementary angles.

Theorem 9: If two adjacent angles have their noncommon sides lying on a line, then they are supplementary angles.

Supplementary angles do not need to be adjacent (Figure 8

**Figure 8 **Nonadjacent supplementary angles.

Because *m* ∠8 + *m* ∠9 = 180°, ∠8 and ∠9 are supplementary.

*Theorem 10:* If two angles are supplementary to the same angle, or to equal angles, then they are equal to each other.