ICSE Class 8 Maths Geometry Chapter 01 Fundamental Concepts

Fundamental Concepts

Lines and Angles

You are familiar with geometrical objects such as points, straight lines, rays, line segments, angles and parallel lines. In this chapter, we will revise what you have already learnt.

Point

A point has a position but it has no length, width or thickness. It is represented by a dot on a sheet of paper.

Line

A straight line or a line is formed by a collection of points. Its basic quality is straightness. A line has position and shape, but not breadth or thickness. A line has no end points and can be extended indefinitely in both directions, so it does not have a definite length.

Ray

A ray is a part of a straight line. It extends indefinitely in one direction from a fixed point. Thus, a ray has one end point and it does not have a definite length.

Line segment

A line segment is a part of a straight line between two points. It has two end points and a finite length.

Plane

A plane is a flat surface that has length and breadth but no thickness.

Angle

Two rays or line segments starting from the same point, form an angle. The common end point is called the vertex of the angle. The rays or line segments are called the arms of the angle. Angles are usually measured in degrees (\(^\circ\)) and are given names in accordance with their magnitudes, as shown in the following figures.

Types of Angles

Acute angle (between 0° and 90°): An angle that measures between 0 and 90 degrees.

Right angle (90°): An angle that measures exactly 90 degrees.

Obtuse angle (between 90° and 180°): An angle that measures between 90 and 180 degrees.

Straight angle (180°): An angle that measures exactly 180 degrees.

Reflex angle (between 180° and 360°): An angle that measures between 180 and 360 degrees.

Adjacent angles

Two angles are called adjacent angles if they have the same vertex and a common arm and their other arms are on either side of the common arm. In the figure, \(\angle POQ\) and \(\angle QOR\) are adjacent angles, while \(\angle POR\) and \(\angle POQ\) are not adjacent angles.

Linear pair

Two adjacent angles are said to form a linear pair if the sum of their measures is 180°. Here, \(\angle AOB\) and \(\angle AOC\) form a linear pair because \(115° + 65° = 180°\).

Property: If a ray OA stands on a line BC then the adjacent angles \(\angle AOB\) and \(\angle AOC\) form a linear pair, that is, \(\angle AOB + \angle AOC = 180°\).

Also, the sum of all the angles at a point of a line on one side of it is 180°. Here, \(\angle BOQ + \angle QOP + \angle POC = 180°\).

Complementary angles

Two angles are called complementary angles if the sum of their measures is 90°. Each angle is said to be the complement of the other. For example, two angles of measures 32° and 58° are complementary angles.

Supplementary angles

Two angles are called supplementary angles if the sum of their measures is 180°. Each angle is said to be the supplement of the other. For example, angles of measures 104° and 76° are supplementary angles.

Angles at a point

The sum of all the angles at a point, making a complete rotation, is 360°. In the figure, \(60° + 90° + 90° + 120° = 360°\).

Vertically opposite angles

When two straight lines AB and CD intersect at the point O then the pairs (i) \(\angle AOD\) and \(\angle COB\) and (ii) \(\angle AOC\) and \(\angle BOD\) are called vertically opposite angles.

Property: If two lines intersect then the vertically opposite angles so formed are equal. In the figure, \(\angle AOD = \angle COB\) and \(\angle AOC = \angle BOD\).

Solved Examples

Example 1 If an angle is \(\frac{2}{3}\) of its supplement, find the angles.

Solution Let the angle \(= x°\). So, its supplement \(= 180° - x°\).

Given that \(x° = \frac{2}{3}(180° - x°) \Rightarrow 3x° = 2(180° - x°) \Rightarrow 3x° = 360° - 2x°\)

\(\Rightarrow 3x° + 2x° = 360° \Rightarrow 5x° = 360° \Rightarrow x° = \frac{360°}{5} = 72°\).

So, the angles are 72° and 180° - 72°, i.e., 108°.

Example 2 Find the values of a and b from the adjoining figure, where POQ is a straight line.

Solution Here, OT stands on the line PQ. So, \(\angle POT + \angle QOT = 180°\)

\(\Rightarrow 3a + 22° + 47° = 180° \Rightarrow 3a = 180° - 22° - 47° = 111°\).

\(\therefore a = \frac{111°}{3} = 37°\).

Being angles at a point on one side of the line PQ, \(\angle POR + \angle ROS + \angle SOQ = 180°\)

\(\Rightarrow 90° + a + b + 18° = 180° \Rightarrow 90° + 37° + b + 18° = 180°\).

\(\therefore b = 180° - 145° = 35°\).

Hence, \(a = 37°\) and \(b = 35°\).

Example 3 Find the values of a and b from the adjoining figure when \(a - b = 4°\).

Solution \(\angle COE =\) vertically opposite \(\angle DOF = 90°\).

\(\therefore\) sum of the angles at a point on one side of a straight line \(= 180°\); \(\angle AOE + \angle COE + \angle BOC + \angle BOC = 180°\)

\(\Rightarrow a + 22° + 90° + b + 36° = 180°\)

\(\Rightarrow a + b + 148° = 180° \Rightarrow a + b = 32°\) ... (1)

Given that \(a - b = 4°\) ... (2)

Adding (1) and (2), we get \(2a = 36° \Rightarrow a = 18°\).

From (1), \(b = 32° - a = 32° - 18° = 14°\).

Hence, \(a = 18°\), \(b = 14°\).

Example 4 Find the values of a, b and c from the adjoining figure, where \(a : b : c = 1 : 2 : 3\).

Solution Let \(a = x\), \(b = 2x\) and \(c = 3x\).

Then \(2c + 3b - 3a = 6x + 6x - 3x = 9x\), \(5c = 15x\), \(3b = 6x\), \(2a + b = 2x + 2x = 4x\) and \(2a = 2x\).

\(\therefore\) sum of the angles at a point \(= 360°\),

\(9x + 15x + 6x + 4x + 2x = 360° \Rightarrow 36x = 360° \Rightarrow x = \frac{360°}{36} = 10°\).

Hence, \(a = 10°\), \(b = 20°\) and \(c = 30°\).

Teacher's Note

Understanding angles helps us navigate the world - from setting compass directions to designing buildings and programming robotics.

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