ICSE Class 8 Maths Geometry Chapter 25 Points and Planes Lines and Angles

Unit Five: Geometry

Points and Planes, Lines and Angles

Triangles

Construction of Triangles

Quadrilaterals

Construction of Quadrilaterals

Polygons

Symmetry, Reflection and Rotation

Let's Recap

1. Find the complement of angle A = 57°.

2. Find the supplement of angle B = 87°.

3. Find the measure of reflex angle ABC, given angle CBA = 93°.

4. Is triangle ABC with sides AB = 9.33 cm, BC = 2.66 cm and CA = 6.66 cm possible?

5. Is triangle ABC with angle ABC = x + 30°, angle BCA = 2x + 60° and angle CAB = 3x + 90° possible?

6. Find the value of x in the following figures:

(i) An angle marked x adjacent to an angle of 36°

(ii) Multiple angles marked 2x, x, 9x, and other values

A comic dialogue shows students asking about calculating tape needed for a kite at a school annual day event, with a teacher explaining the Pythagorean Theorem and introducing geometry concepts.

25 - Points and Planes, Lines and Angles

Points and Planes

Geometry is the study of position, size, and shape of points, planes, lines, and angles.

A point is a small mark that has position, but no magnitude. It has neither shape nor size. On paper, a point is represented by a small dot and denoted by a capital letter from the English alphabet, like point A, point B, and point C.

A flat surface that extends indefinitely in all directions is known as a plane.

Now, to relate a point to a plane, think of a single seed inside a big watermelon as a point. In how many ways can a knife cut through the watermelon so as to cut through that seed? Yes, there can be an infinite number of ways.

This is because a single point can lie on an infinite number of planes. Similarly, two points can lie on an infinite number of planes. But three points can lie on one plane only.

The line segment that joins two points on a plane is the shortest distance between them. If the line segment is extended in both directions endlessly, we realize that two points on a plane can be connected by one and only one line passing through them.

In geometry the word space is used to describe all points, lines, and planes that lie in the entire universe.

A single point can lie on an infinite number of planes.

Two points can lie on an infinite number of planes.

Three or more points can lie on only one plane.

Angles

Two lines on the same plane may intersect or be parallel to each other. An angle is formed when two lines intersect. In Figure 25.1, lines l and m intersect at point O to form angle AOC. OA and OC are the arms of the angle while point O is known as its vertex. The distance arm OC will have to cover, on being rotated, to be in line with arm OA is known as the magnitude of angle AOC. The magnitude of an angle is measured in degrees. A full sweep of one arm of an angle measures 360° or a complete angle.

For any angle x, magnitude of:

a zero angle: \(\angle x = 0°\)

an acute angle: \(0° < \angle x < 90°\)

a right angle: \(\angle x = 90°\)

an obtuse angle: \(90° < \angle x < 180°\)

a straight angle: \(\angle x = 180°\)

a reflex angle: \(180° < \angle x < 360°\)

a complete angle: \(\angle x = 360°\)

The sum of angles about a point is thus 360°. In Figure 25.2, angles 1, 2, 3, 4, and 5 are all angles about the same vertex O.

But \(\angle 1 + \angle 2 + \angle 3 + \angle 4 + \angle 5 = \angle AOA = \text{complete angle} = 360°\)

As the measure of a complete angle = 360°,

\(\angle 1 + \angle 2 + \angle 3 + \angle 4 + \angle 5 = 360°\)

Special Pairs of Angles - Complementary Angles

When the sum of two angles is 90°, they are said to be complementary angles and each angle is known as a complement of the other.

In Figure 25.3 (i) and (ii), angle AOB and angle BOC are complementary angles.

Supplementary Angles

When the sum of two angles is 180°, they are said to be supplementary angles and each angle is known as a supplement of the other.

In Figure 25.4 (i) and (ii), angle DOE and angle EOF are supplementary angles.

Try this! Find the angle which measures twice its complement.

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