Angles are formed when two lines intersect each other at a point. The pair of angles are nothing but two angles. The angle pairs can relate to each other in various ways. Those are complementary angles, supplementary angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, and corresponding angles.

## Pairs of Angles – Definition

The region between two infinitely long lines pointing a certain direction from a common vertex is called an angle. Which is the amount of turn is measured by an angle. The pairs of angles mean two angles. If there is one common line for two angles, it is known as angle pairs. Get the definitions and examples of all pairs of angles in the following section.

### 1. Complementary Angles

Two angles whose sum is 90Â° are called complementary angles and one angle is the complement of another angle.

Here, âˆ AOB = 20Â°, âˆ BOC = 70Â°

So, âˆ AOB + âˆ BOC = 20Â° + 70Â° = 90Â°

Therefore, âˆ AOB and âˆ BOC are called complementary angles.

âˆ AOB is a complement of âˆ BOC and âˆ BOC is the complement of âˆ AOB.

**Example:**

(i) Angles of measure 50Â° and 40Â° are complementary angles because 50Â° + 40Â° = 90Â°.

Thus, the complementary angle of 50Â° is the angle measure 40Â°. The complementary angle of 40Â° is the angle measure 50Â°.

(ii) Complementary of 60Â° is 90Â° – 60Â° = 30Â°

(iii) Complementary of 45Â° is 90Â° – 45Â°= 45Â°

(vi) Complementary of 25Â° is 90 – 25Â° = 65Â°

**Working rule**: To find the complementary angle of a given angle subtracts the measure of an angle from 90Â°.

So, the complementary angle = 90Â° – given angle

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### 2. Supplementary Angles

The pair of angles whose sum is 180Â° is called the supplementary angles and one angle is called the supplement of the other angle.

Here, âˆ AOC = 120Â°, âˆ COB = 60Â°

âˆ AOC + âˆ COB = 120Â° + 60Â° = 180Â°

Therefore, âˆ AOC, âˆ COB are called supplementary angles.

âˆ AOC is the supplement of âˆ COB, âˆ COB is a supplement of âˆ AOC.

**Example:**

(i) Angles of measure 90Â° and 90Â° are supplementary angles because 90Â°+ 90Â° = 180Â°

Thus, the supplementary angle of 90Â° is the angle of measure 90Â°.

(ii) Supplement of 100Â° is 180Â° – 100Â° = 80Â°

(iii) Supplement of 50Â° is 180Â° – 50Â° = 130Â°

(iv) Supplement of 95Â° is 180Â° – 95Â°= 85Â°

(v) Supplement of 140Â° is 180Â°- 140Â° = 40Â°

**Working rule**: To find the supplementary angle of the given angle, subtract the measure of angle from 180Â°.

So, the supplementary angle = 180Â° – given angle

### 3. Adjacent Angles

Two non-overlapping angles are said to be adjacent angles if they have a common vertex, common arm, and other two arms lying on the opposite side of this common arm so that their interiors do not overlap.

In the above figure, âˆ DBC and âˆ CBA are non-overlapping, have BC as the common arm and B as the common vertex. The other arms BD, AB of the angles âˆ DBC and âˆ CBA are opposite sides of the common arm BC.

Hence, the arm âˆ DBC and âˆ CBA form a pair of adjacent angles.

### 4. Linear Pair of Angles

The angles are called liner pairs of angles when they are adjacent to each other after the intersection of two lines. Two adjacent angles are said to form a linear pair if their sum is 180Â°. The types of linear pairs of angles are alternate exterior angles, alternate interior angles, and corresponding angles.

**Alternate interior angles**

Two angles in the interior of the parallel lines and on opposite sides of the transversal. Alternate interior angles are non-adjacent and congruent.

**Alternate exterior angles**

Two angles in the exterior of the parallel lines, and on the opposite sides of the transversal. Alternate exterior angles are non-adjacent and congruent.

**Corresponding angles**

The pair of angles, one in the interior and another in the exterior that is on the same side of the transversal. Corresponding angles are non-adjacent and congruent.

### 5. Vertical Angles

Two angles formed by two intersecting lines having no common arm are called the vertically opposite angles.

When two lines intersect, then vertically opposite angles are always equal.

âˆ 1 = âˆ 3

âˆ 2 = âˆ 4

### Pair of Angles Examples

**Example 1:**

Suppose two angles âˆ AOC and âˆ BOC form a linear pair at point O in a line segment AB. If the difference between the two angles is 40Â°. Then find both the angles.

**Solution:**

Given that,

âˆ AOC and âˆ BOC form a linear pair

so, âˆ AOC + âˆ BOC = 180Â° —- (i)

âˆ AOC – âˆ BOC = 40Â° —- (ii)

Add both equations

âˆ AOC + âˆ BOC + âˆ AOC – âˆ BOC = 180Â° + 40Â°

2âˆ AOC = 220Â°

âˆ AOC = 220Â° / 2

âˆ AOC = 110Â°

Now, substitute âˆ AOC in (i)

110Â° + âˆ BOC = 180Â°

âˆ BOC = 180Â° – 110Â°

âˆ BOC = 70Â°

Therefore, two angles are 70Â°, 110Â°.

**Example 2:**

Find the values of the angles x, y, and z in the following figure.

**Solution:**

From the given figure,

lines AD and EC intersect each other and âˆ DOC and âˆ AOE are vertically opposite angles

When two lines intersect, then vertically opposite angles are always equal.

So, âˆ DOC = âˆ AOE

Therefore, z = 40Â°

AD is a line

âˆ DOE and âˆ AOE are adjacent angles. The sum of adjacent angles are 180Â°

So, âˆ DOE + âˆ AOE = 180Â°

y + 40Â° = 180Â°

y = 180Â°- 40Â°

y = 140Â°

And, lines AD and CE intersect

âˆ DOE, âˆ COA are vertically opposite.

When two lines intersect, then vertically opposite angles are always equal.

So, âˆ DOE = âˆ COA

y = âˆ COB + âˆ BOAA

140Â° = x + 25Â°

140Â° – 25Â° = x

x = 115Â°

Hence, x = 115Â°, y = 140Â°, z = 40Â°

**Example 3:**

Identify the five pairs of adjacent angles in the following figure.

**Solution:**

Adjacent angles are the angles that have a common side, vertex, and no overlap.

So, (i) âˆ AOD, âˆ AOE are the adjacent angles

The common side is AO, the common vertex is O and OE, OD is not overlapping.

(ii) âˆ AOD, âˆ DOB is the adjacent angles

The common vertex is O, the common side is OD and OA, OB is not overlapping.

(iii) âˆ DOB, âˆ BOC is the adjacent angles

The common side is OB, the common vertex is O, and OD, OC are not overlapping.

(iv) âˆ COE, âˆ BOC are adjacent angles

The common side is OC, the common vertex is O, and OE, OB are not overlapping.

(v) âˆ COE, âˆ EOA are adjacent angles

The common vertex is O, the common side is OE, and OC, OA is not overlapping.