ICSE Class 6 Maths Chapter 17 Angles

Chapter 17

Angles

(With their Types and Properties)

Concept Of An Angle

Two different rays starting from the same fixed point form an angle.

In the adjoining figure, two different rays OA and OB start from the same fixed point O to form angle AOB.

The point O, which is common to both the rays, is called the vertex of the angle AOB, whereas the rays OA and OB are called the sides or arms of the angle AOB.

The symbol '\(\angle\)' is used to represent an angle.

Thus, angle AOB can be written as \(\angle AOB\), i.e. angle AOB = \(\angle AOB\).

An angle is represented by three capital letters taken in such a way that the letter in the middle is always the vertex of the angle.

For example:

(i) \(\angle ABC\) or \(\angle CBA\)

(ii) \(\angle PQR\) or \(\angle RQP\)

An angle can be represented by the name of its vertex also. Thus, \(\angle ABC = \angle CBA = \angle B\); \(\angle PQR = \angle RQP = \angle Q\) and so on.

Sometimes, two line segments with a common end point also form an angle at that point.

In the adjoining figure, line segments BA and BC form angle ABC at their common end point.

It should be noted here that, whether or not BA and BC are rays or line segments, the measure of angle ABC remains the same.

If the arms of an angle have several points on them, the same angle can be written in various ways.

In the given figure: \(\angle O = \angle AOB = \angle BOA = \angle COB = \angle BOD = \angle BOC = \angle DOA = \angle DOC = \angle AOD\)

Interior Of An Angle

The interior of an angle is the region that lies within an angle. In other words, it is the region bounded by the arms of an angle.

The shaded portion of the given figure shows the interior of the angle AOB.

Teacher's Note

Understanding angle components helps students recognize angles in real objects like door hinges, clock hands, and roof slopes in everyday architecture.

Exterior Of An Angle

The exterior of an angle is the region that lies outside the angle. The shaded portion of the adjoining figure shows the exterior of the angle AOB.

Angle Formed By Rotation

Consider a ray OA rotating about its end point O (OA is the ray's original or initial position). As OA rotates about the fixed point O, it attains several positions before it returns to its original position.

At any particular moment, the position of ray OA, shown above as OA', is termed its final position. It can be observed easily that the initial position OA and the final position OA' form an angle \(\angle AOA'\). The angle becomes bigger and bigger as OA' rotates till it coincides with its initial position OA (see the following figures carefully).

The figures show various stages of ray rotation:

(i) First position - Initial position

(ii) Final position - Initial position

(iii) Counterclockwise - Initial position

(iv) Final position - Initial position

(v) Final position - Initial position

(vi) Final position - Initial position

(vii) Initial position - Final position

(viii) (OA' coinciding with OA) - Initial position = Final position

Teacher's Note

Rotational angles appear when observing clock hands moving throughout the day or when watching windmill blades rotate, making this concept directly observable in daily life.

Measuring An Angle

The unit of measurement of an angle is degree. The symbol for degree is °.

Thus, if measure of an angle is 60 degrees, we write 60°.

The angle measure of one complete rotation is 360 degrees i.e. 360° (see alongside)

When the angle formed by one complete rotation of a ray is divided into 360 equal parts, each part is called one degree, i.e. 1°.

If one degree is further divided into 60 equal parts, each part is called a minute, which is denoted by 1' (one prime).

Thus, 1 minute = 1', 5 minutes = 5', 40 minutes = 40' and so on.

If we further divide one minute into 60 equal parts, each part is called a second, which is denoted by 1" (two primes).

Thus, 1 second = 1", 45 seconds = 45", 40 seconds = 40" and so on.

Hence, 1 complete rotation = 360° (Three sixty degrees)

1° = 60' (Sixty minutes)

1' = 60" (Sixty seconds)

5 minutes 30 seconds = 5' 30"

25 degrees 30 minutes 15 seconds = 25° 30' 15" and so on.

Example 1:

Add: (i) 32° 23' 15" and 49° 17' 32" (ii) 74° 35' 18" and 9° 20' 53"

Solution:

(i) 32° 23' 15"

+ 49° 17' 32"

= 81° 40' 47" (Ans.)

(ii) 74° 35' 18"

+ 9° 20' 53"

= 83° 55' 71"

Since 60" = 1' ∴ 71" = 1' 11" and so 83° 55' 71" = 83° 56' 11" (Ans.)

Using A Protractor For Measuring An Angle

A protractor, as shown below, is a semi-circular plastic (or metallic) disc marked in degrees from 0° to 180° on its semi-circular part. The centre of this semi-circular piece is marked as O, which is also the mid-point of its base-line.

In order to measure an angle, say angle PQR, as shown above, the base line of the protractor is kept on arm QR of the angle, and its centre O is kept on the vertex of the angle the PQR. Now the position of the other arm, i.e. arm PQ, of the angle is read from the markings on the protractor.

In the figure given above \(\angle PQR = 40°\)

Teacher's Note

Using a protractor to measure angles in triangles and polygons connects geometry to practical applications like construction, design, and engineering.

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