ICSE Class 8 Maths Chapter 23 Fundamental Concepts

Unit 4: Geometry - Chapter 23: Fundamental Concepts

23.1 Review

Teacher's Note

Points and lines are fundamental building blocks in geometry that help us describe the physical world around us, from the corners of a building to the edges of a road.

23.2 Types of Angles

1. Acute angle: It is the angle which measures between 0° and 90°.

Examples shown: 75°, 42°, 25°, etc.

2. Right angle: It is the angle which measures 90°.

Examples shown with right angle symbols at 90°.

3. Obtuse angle: It is the angle which measures between 90° and 180°.

Examples shown: 150°, 120°, 140°, etc.

4. Straight angle: It is the angle which measures 180°.

Examples shown: 180° angles.

5. Reflex angle: It is the angle which measures between 180° and 360°.

Examples shown: 270°, 240°, 260°, etc.

Teacher's Note

Understanding different types of angles helps us measure and construct various shapes we encounter daily, from the corners of boxes to the angles in building architecture.

23.3 More About Angles

1. Angles on a point: Whatever be the number of angles formed at a point, their sum is always 360°.

i.e., \(\angle AOB + \angle BOC + \angle COD + \angle DOE + \angle EOA = 360°\)

2. Angles on the same side of a straight line: Whatever be the number of angles formed at a point of a straight line on the same side of it, their sum is always 180°.

i.e., \(\angle AOB + \angle BOC + \angle COD + \angle DOE = 180°\)

3. Adjacent angles: Two angles are said to be adjacent angles if they have a common vertex, a common arm and the other arms of the two angles lie on the opposite sides of the common arm.

If the sum of two adjacent angles is 180°, their outer arms are in a straight line. Such adjacent angles are said to form a linear pair of angles.

4. Vertically opposite angles: When two 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.

Vertically opposite angles are always equal.

\(\therefore \angle AOD = \angle BOC\) and \(\angle AOC = \angle BOD\)

5. Complementary angles: Two angles are said to be complementary, if their sum is one right angle i.e. 90°. Each angle is called complement of the other.

6. Supplementary angles: Two angles are said to be supplementary, if their sum is two right angles i.e. 180°. Each angle is called supplement of the other.

Teacher's Note

These angle relationships are essential for understanding how objects intersect and balance, such as how roof beams meet at complementary angles to form stable structures.

Test Yourself

1. In a linear pair of adjacent angles, one angle is 110°, then the other angle is ...

2. Two adjacent angles are 90° each, they form a ...

3. Two vertically opposite angles are (x - 20)° and 85°; then ... ... = ... and x = ...

4. Complement of 80° = ... and its supplement = ...

5. Supplement of 70° = ... and its complement = ...

Example 1

In the given figure, AB is a straight line. Find the values of x and y.

Solution:

Since, AB is a straight line and the sum of the angles at a point on the same side of straight line is 180°.

\(\therefore 5x + 10° + 65° = 180° \Rightarrow 5x = 180° - 75°\)

i.e., \(5x = 105° \Rightarrow x = \frac{105°}{5} = 21°\)

For the same reason:

\(2x + 90° + 3y = 180° \Rightarrow 2 \times 21° + 90° + 3y = 180°\)

i.e., \(132° + 3y = 180° \Rightarrow 3y = 180° - 132° = 48°\)

i.e., \(y = \frac{48°}{3} = 16°\)

\(\therefore x = 21°\) and \(y = 16°\) (Ans.)

Example 2

The supplement of an angle is 10° more than three times its complement. Find the angle.

Solution:

Let the angle be x.

\(\therefore\) Its supplement = 180° - x and its complement = 90° - x

Given: \(180° - x = 3(90° - x) + 10°\)

\(\Rightarrow 180° - x = 270° - 3x + 10° i.e., 3x - x = 280° - 180°\)

\(\Rightarrow 2x = 100° \text{ and } x = \frac{100°}{2} = 50°\) (Ans.)

Example 3

x and y form a linear pair of two adjacent angles. If y = 3x - 12°; find the values of x and y.

Solution:

Since, x and y form a linear pair of two adjacent angles,

\(x + y = 180°\)

\(\Rightarrow x + 3x - 12° = 180°\) [\(\therefore y = 3x - 12°\)]

\(\therefore 4x = 180° + 12° \Rightarrow 4x = 192° \text{ and } x = \frac{192°}{4} = 48°\)

\(x + y = 180° \Rightarrow 48° + y = 180° i.e. y = 180° - 48° = 132°\)

\(\therefore x = 48°\) and \(y = 132°\) (Ans.)

Test Yourself

6. In the given figure, x + y = ..., if AOB is ...

7. The adjacent angles on a straight line are in the ratio 7 : 5; the angles are ... and ... .

8. If x + 10° and 2x - 40° are complementary \(\Rightarrow\) ... = 90° and x = ... .

9. If 3x + 40° and 5x + 20° are supplementary, \(\Rightarrow\) ... = 180° and x = ... .

10. The angle between bisectors of two adjacent complementary angles is ... .

11. The angle between bisectors of two adjacent supplementary angles is ... .

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