Angles and Pairs of Angles

Identification of Regions and Points That Lie Inside, Outside, and On the Angles

Rohit and Mohit start walking from the same point in different directions. Let Rohit move towards Y and Mohit towards Z as shown below.

Here, we can say that the rays form an angle ZXY. These rays, i.e., are known as the arms of ∠ZXY. The point, i.e. X, that is common to these arms is called the vertex of ∠ZXY.

Let us consider the following ∠CAB with some points P, Q, and R.

Let us shade the different parts of the angle as shown below.

Here, we can see that the region of the angle shaded by yellow colour lies between the two arms of the angle. This region is called the interior region of the angle. It extends indefinitely, since the two arms also extends indefinitely. Every point in this region is said to lie in the interior of the angle. In this figure, point P lies in the interior of ∠CAB.

The region of the angle shaded pink lies outside the two arms of the angle. This region is called the exterior region of the angle. Like the interior region of an angle, the exterior region also extends indefinitely. Every point in this region is said to lie in the exterior of the angle. In this figure, point R lies in the exterior of ∠CAB.

The boundary of ∠CAB is formed by its arms. These arms are called the boundary of the angle. Every point lying on the arms is said to lie on the boundary of the angle, or simply, on the angle. In this figure, points A, B, C, and Q lie on the angle.

Using this concept, we can say that an angle has three regions. They are interior region, exterior region, and boundary region. Using this idea, we can easily identify whether a point lies inside, outside, or on the given angle. Let us discuss one more example to understand the concept better.

Example 1:

In the figure given below, name the point or points that lie

(i) in the interior of the angle

(ii) in the exterior of the angle

(iii) on the angle

Solution:

(i) Points B and E lie in the interior of the angle.

(ii) Points C and D lie in the exterior of the angle.

(iii) Points A, F, X, Y, and Z lie on the angle.

In some cases, pairs of angles show some special properties. Complementary and supplementary angles are examples of such pairs of angles, whose sum of measures exhibit a relationship. Go through the video to learn more about them.

So, the two important definitions of complementary and supplementary angles are as follows.

“If the sum of the measures of two angles is 90°, then the two angles are said to be complement to each other or complementary angles”.

“If the sum of the measures of two angles is 180°, then the two angles are said to be supplement to each other or supplementary angles.”

Let us solve some examples related to complementary and supplementary angles to understand the concept better.

Example 1:

Find the complement of the following angles.

52° and 75°

Solution:

Complement of 52° = 90° − 52°

= 38°

Complement of 75° = 90° − 75°

= 15°

Example 2:

Find the supplement of the following angles.

100° and 36°

Solution:

Supplement of 100° = 180° − 100°

= 80°

Supplement of 36° = 180° − 36°

= 144°

Example 3:

Can two acute angles be supplementary angles?

Solution:

No, two acute angles cannot be supplementary angles. The measure of an acute angle is less than 90°. Therefore, the sum of the measures of two acute angles is always less than 180°.

Example 4:

Write True or False.

(i) The opposite angles of a square are complementary angles.

(ii) Two obtuse angles can be supplementary angles.

Solution:

(i) False, as each angle of a square is a right angle.

Sum of two opposite angles = 90° + 90°

= 180°

(ii) False, because the measure of an obtuse angle is greater than 90°. Therefore, the sum of the measures of two obtuse angles cannot be 180°.

Example 5:

An angle measures four times its supplementary angle. Find the measures of both the angles.

Solution:

Let the measure of the supplementary angle of the given angle be x° then the measure of the given angle will be 4x°.     
According to the definition of supplementary angles, we obtain

4x° + x° = 180° 

⇒ 5x° = 180° 

⇒ x° = 36°

∴ 4x° = 4 × 36°

⇒ 4x° = 144° 

Thus, the measures of the given angles are 144° and 36°.

Example 6:

The measure of an angle is 6° more than the twice of its complementary a…