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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.

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.

A ray OZ stands on a line XY such that ∠ZOY = 72°, as shown in the following figure.

Can we find ∠XOZ?

We can find ∠XOZ using the concept of linear pair.

Therefore, first of all let us know about the linear pair of angles.

A linear pair is a pair of adjacent angles and whose non-common sides are opposite rays…

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