ICSE Class 7 Maths Chapter 23 Lines and Angles

Unit 4 - Geometry

Chapter 23 - Lines and Angles

(Including Construction of Angles)

23.1 Review

1. Point

A point is a mark of position, which has no length, no breadth and no thickness. In general, it is represented by a capital letter as shown alongside.

2. Line

A line has length, but no breadth or thickness. The given figure shows a line AB in which two arrow-heads in opposite directions show that it can be extended infinitely in both the directions.

A line may be straight or curved but when we say 'a line' it means a straight line only.

Each line, whatever be its length, has an infinite number of points in it.

3. Ray

It is a straight line which starts from a fixed point and moves in the same direction. The given figure shows a ray AB with fixed initial point A and moving in the direction AB.

4. Line Segment

It is a straight line with its both ends fixed. The given figure shows a line segment, whose both the ends A and B are fixed.

The adjoining figure shows a line AB which can be extended upto infinity on its both sides.

The adjoining figure shows a ray AB with fixed end as point A and which can be extended upto infinity through point B. It is clear from the figure, that a ray is a part of a line.

The adjoining figure shows a line-segment AB with fixed ends A and B. It is clear from the different figures, that a line-segment is a part of a ray as well as of a line. Also, a line segment is the shortest distance between two fixed points.

5. Angle

An angle is formed when two line segments or two rays have a common end-point. The two line segments, forming an angle, are called the arms of the angle whereas their common end-point is called the vertex of the angle.

The adjacent figure represents an angle ABC or ∠ABC or simply ∠B. AB and BC are the arms of the angle and their common point B is the vertex.

Teacher's Note

Understanding points, lines, rays, and segments is fundamental to geometry - you encounter these concepts when drawing maps, designing buildings, or even arranging furniture in a room.

23.2 Measurement of an Angle

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

Thus: 60 degree = 60°, 87 degree = 87° and so on.

If one degree is divided into 60 equal parts, each part is called a minute (') and if one minute is further divided into 60 equal parts, each part is called a second ("").

Thus,

(i) 1° = 60' and 1' = 60"

(ii) 8 minutes 45 seconds = 8' 45"

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

23.3 Types of Angles

Acute angle: measures less than 90°

Right angle: measures 90°

Obtuse angle: measures between 90° and 180°

Straight angle: measures 180°

Reflex angle: measures between 180° and 360°

23.4 More About Angles

1. Angles about a point: If a number of angles are formed about a point, their sum is always 360°.

In the adjoining figure:

∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA = 360°.

2. Adjacent angles: Two angles are said to be adjacent angles, if:

(i) they have a common vertex,

(ii) they have a common arm and

(iii) the other arms of the two angles lie on opposite sides of the common arm.

The adjoining figure shows a pair of adjacent angles: because (i) they have a common vertex (O), (ii) they have a common arm (OB) and (iii) the other arms OA and OC of the two angles are on opposite sides of the common arm OB.

Teacher's Note

Angles are everywhere in daily life - from the angle at which you tilt your head to the angles in building architecture and sports activities.

3. Vertically Opposite Angles

When two straight lines intersect each other four angles are formed. The pair of angles which lie on the opposite sides of the point of intersection are called vertically opposite angles.

In the adjoining figure, two straight lines AB and CD intersect each other at point O. Angles AOD and BOC form one pair of vertically opposite angles, whereas angles AOC and BOD form another pair of vertically opposite angles.

Vertically opposite angles are always equal.

i.e. ∠AOD = ∠BOC and ∠AOC = ∠BOD.

Important: In the adjoining figure, rays OX and OY meet at O to form ∠XOY (i.e., ∠a) and reflex ∠XOY (i.e., ∠b). It must be noted that ∠XOY represents the smaller angle only, unless it is mentioned otherwise.

23.5 Complementary and Supplementary Angles

1. Two angles are called complementary angles, if their sum is one right angle, i.e., 90°. Each angle is called the complement of the other.

e.g., 20° and 70° are complementary angles, because 20° + 70° = 90°.

Clearly, 20° is the complement of 70° and 70° is the complement of 20°.

Thus, the complement of angle 53° = 90° - 53° = 37°.

2. Two angles are called supplementary angles, if their sum is two right angles, i.e., 180°. Each angle is called the supplement of the other.

e.g., 30° and 150° are supplementary angles because 30° + 150° = 180°.

Clearly, 30° is the supplement of 150° and vice-versa.

Thus, the supplement of 105° = 180° - 105° = 75°.

Exercise 23 (A)

1. State true or false:

(i) A line segment 4 cm long can have only 2000 points in it.

(ii) A ray has one end point and a line segment has two end-points.

(iii) A line segment is the shortest distance between any two given points.

(iv) An infinite number of straight lines can be drawn through a given point.

(v) 40° is the complement of 60°.

(vi) 45° is the supplement of 45°.

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