ICSE Class 8 Maths Chapter 21 Fundamental Geometrical Concepts

Chapter 21: Fundamental Geometrical Concepts

In this chapter, we shall have a brief revision of geometrical concepts, angles, their types and properties of angles associated with parallel lines. Stress is being laid on their applications and numerical problems.

Geometrical Concepts

Point

A small dot marked by a sharp pencil on a sheet of paper or a prick made by a fine needle on a paper are examples of a point.

A point determines a location in space. It has no length, breadth or thickness.

Line

A line has length only. It has no breadth or thickness.

The basic concept of a line is its straightness and it extends indefinitely in both directions. The two arrowheads in the opposite directions indicate that the length of a line is unlimited i.e. it has no definite length.

A line has no end points and it consists of an infinite (uncountable) number of points.

Line Segment

A line segment is a portion of a line. It has two end points and a definite length.

Ray

A ray is a part of a line that extends indefinitely in one direction from a point on the line, say O. O is the initial point of the ray.

A ray has only one end point and it has unlimited length.

Plane

A plane has length and breadth. It has no thickness.

The basic concept of a plane is its flatness and it extends indefinitely in all directions. Thus, a plane is a flat surface which extends indefinitely in all directions. The length and breadth of a plane are unlimited i.e. a plane has no definite length and no definite breadth.

An unlimited number of lines can be drawn in a plane through a given point in a plane.

Exactly one line passes through two different given points in a plane and it lies wholly in that plane.

Two different lines in a plane either intersect at exactly one point or are parallel.

Teacher's Note

Points and lines form the foundation of geometry - we see them everywhere from the corners of a room (point) to the edges of a table (line segment).

Space

Space is the set of all points in the universe. So, a point or a line or a plane is a subset of space. In fact, everything we look at is a part of space.

Axiom

The statements which are obviously true are called axioms (or postulates) and are accepted without proof. In fact, these form the basis of the subject.

Theorem

A theorem is a statement of mathematical truth which has to be proved from the facts already proved or assumed.

Corollary

A statement whose truth can easily be derived from a theorem is called its corollary.

Angle

An angle is the figure formed by two rays (or line segments) with same initial (or common) point.

The initial (or common) point is its vertex and the two rays (or line segments) are its arms or sides.

Types Of Angles

Acute angle - an angle whose measure lies between 0° and 90°.

Right angle - an angle whose measure is 90°. It is a quarter turn.

Obtuse angle - an angle whose measure lies between 90° and 180°.

Straight angle - an angle whose measure is 180°. It is a half turn.

Reflex angle - an angle whose measure lies between 180° and 360°.

Teacher's Note

Angles help us understand directions and rotations - from the angle of a ladder against a wall to the turn of a steering wheel.

Adjacent Angles

Two angles are called adjacent angles if they have a common vertex, a common arm and their other arms lie on either side of the common arm.

In the adjoining figure, angle AOB and angle BOC are adjacent angles.

If the sum of any two adjacent angles is 180° then their exterior arms are in the same straight line.

In the adjoining figure, angle AOB + angle BOC = 30° + 150° = 180°. So COA is a straight line.

Linear Pair

Two adjacent angles whose exterior arms are in a straight line are said to form a linear pair i.e. two adjacent angles are said to form a linear pair if the sum of their measures is 180°.

Here, AOB is a straight line, so angle AOB = 180°. Therefore, angle AOC and angle COB form a linear pair.

Complementary Angles

Two angles are called complementary angles if the sum of their measures is 90°. Each angle is called the complement of the other.

Complementary angles need not be adjacent angles.

Supplementary Angles

Two angles are called supplementary angles if the sum of their measures is 180°. Each angle is called the supplement of the other.

Supplementary angles need not be adjacent angles.

Angles At A Point

In the adjoining figure, the four angles together make one complete rotation, so they add upto 360°.

This is true no matter how many angles are formed at a point. Thus:

Sum of angles at a point = 360°

Angles On One Side Of A Straight Line

In the adjoining figure, the three angles together make a straight line, so they add upto 180°.

This is true no matter how many angles make up the straight line. Thus:

Sum of angles at a point on one side of a straight line = 180°

Teacher's Note

When standing at a point and looking in different directions, the total rotation is always 360° - just like a full turn on a spinning chair.

Vertically Opposite Angles

When two straight lines intersect each other, they form four angles, say a, b, c and d.

Angles a and c are called vertically opposite angles (abbreviated vert. opp. angles), and so are angles b and d.

Theorem

If two straight lines intersect, the vertically opposite angles are equal.

Given: Two straight lines AB and CD intersect at the point O.

To prove: angle AOC = angle BOD and angle COB = angle AOD

Proof:

Example 1

An angle is one-fourth of its complement. Find the size of the angles.

Solution:

Let the angle be x°, then its complement is 90° - x°.

According to given information, x° = \(\frac{1}{4}\) (90° - x°)

4x° = 90° - x° ⟹ 5x° = 90°

⟹ x° = 18° and 90° - x° = 90° - 18° = 72°

Hence, the required angles are 18° and 72°.

Example 2

In the adjoining figure, find the values of x and y, given that AOD is a straight line.

Solution:

Since AOD is a straight line and the sum of angles at a point on one side of a straight line is 180°,

85° + 2x + 15° = 180°

⟹ 2x = 180° - 85° - 15° = 80°

⟹ x = 40°

Also x + 90° + y = 180°

⟹ 40° + 90° + y = 180°

⟹ y = 180° - 40° - 90° = 50°

Hence, x = 40° and y = 50°.

Example 3

In the adjoining figure, the three straight lines AB, CD and EF all pass through the point O. If angle BOD = 90° and x : y = 2 : 1, find angle BOE and angle FOD.

Solution:

Given x : y = 2 : 1 ⟹ \(\frac{x}{y}\) = \(\frac{2}{1}\)

⟹ x = 2y

angle AOC = angle BOD (vert. opp. angles)

⟹ angle AOC = 90° (since angle BOD = 90° given)

Since FOE is a straight line, sum of angles at a point on one side of a straight line is 180°.

angle FOA + angle AOC + angle COE = 180°

⟹ y° + 90° + x° = 180°

⟹ y° + 2y° = 180° - 90°

⟹ 3y° = 90° ⟹ y = 30

⟹ x = 2 × 30 = 60

angle BOE = y° (vert. opp. angles)

⟹ angle BOE = 30°

angle FOD = x° (vert. opp. angles)

⟹ angle FOD = 60°.

Example 4

In the given figure, 2b - a = 65 and angle BOC = 90°. Find the measures of (i) angle AOB (ii) angle AOD (iii) angle COD.

Solution:

Since the sum of angles at a point = 360°,

a° + 90° + (2a + b + 15)° + 2b°= 360°

⟹ 3a + 3b = 360 - 90 - 15 = 255

⟹ a + b = 85

Also 2b - a = 65

Adding (i) and (ii), we get

3b = 150 ⟹ b = 50

Substituting this value of b in (i), we get

a + 50 = 85 ⟹ a = 35

(i) angle AOB = a° = 35°

(ii) angle AOD = 2b° = 2 × 50° = 100°

(iii) angle COD = (2a + b + 15)° = (2 × 35 + 50 + 15)° = 135°.

Teacher's Note

Vertically opposite angles are equal - like two mirrors facing each other reflecting the same angle on opposite sides.

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