ICSE Class 9 Maths Chapter 14 Rectilinear Figures

Rectilinear Figures

Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium

Introduction

Rectilinear means along a straight line or in a straight line or forming a straight line.

A plane figure bounded by straight lines is called a rectilinear figure.

A closed plane figure, bounded by at least three line segments, is called a polygon.

Names of Polygons

A polygon is named by the number of sides in it, as given below:

Convex Polygon:

If each angle of a polygon is less than 180 degrees, the polygon is called a convex polygon.

Concave Polygon:

If at least one angle of a polygon is greater than 180 degrees, it is called a concave polygon.

Unless otherwise stated, a polygon means a convex polygon.

In a polygon of n sides, the sum of the interior angles is equal to (2n - 4) right angles.

If the sides of a polygon are produced in order (i.e. all the sides are produced either in clockwise direction or in anti-clockwise direction), the sum of exterior angles so formed is 4 right angles. i.e. ∠a + ∠b + ∠c + ∠d + ... = 4 rt. angles = 360 degrees.

Solved Example 1

The sum of the interior angles of a polygon is five times the sum of its exterior angles. Find the number of sides in the polygon.

Solution:

Let the number of sides be n.

Given: The sum of the interior angles of the polygon = 5 times the sum of its exterior angles.

\[(2n - 4) \times 90° = 5 \times 360°\]

On solving, we get: 2n = 24 and n = 12

The required no. of sides in the polygon = 12

Solved Example 2

One angle of an eight-sided polygon is 100 degrees and the other angles are equal. Find the measure of each equal angle.

Solution:

The sum of the interior angles of an eight-sided polygon

\[= (2n - 4) \times 90° = (2 \times 8 - 4) \times 90° = 1080°\]

Since, one angle of the polygon = 100 degrees

The sum of the remaining seven angles = 1080 degrees - 100 degrees = 980 degrees

Since, these angles are equal

The measure of each equal angle = 980 degrees / 7 = 140 degrees

Alternative method:

Let each of the remaining seven equal angles = x degrees.

The sum of these seven angles = 7x degrees

And, 7x + 100 degrees = (2 times 8 - 4) times 90 degrees, which gives 7x = 1080 degrees - 100 degrees = 980 degrees

x = 980 / 7 = 140 degrees

Solved Example 3

In a pentagon ABCDE, AB is parallel to ED and angle B = 140 degrees. Find the angles C and D, if ∠C : ∠D = 5 : 6.

Solution:

The rough sketch of the given pentagon will be as shown alongside.

Since, AB/ED gives ∠A + ∠E = 180 degrees.

Given: ∠C : ∠D = 5 : 6, which gives if ∠C = 5x, ∠D = 6x.

Now, ∠A + ∠B + ∠C + ∠D + ∠E = (2 times 5 - 4) times 90 degrees

(∠A + ∠E) + 140 degrees + 5x + 6x = 540 degrees

180 degrees + 140 degrees + 5x + 6x = 540 degrees

i.e. 11x = 540 degrees - 320 degrees, which gives 11x = 220 degrees and x = 220 / 11 = 20 degrees

∠C = 5x = 5 times 20 degrees = 100 degrees and ∠D = 6x = 6 times 20 degrees = 120 degrees

Solved Example 4

In the pentagon ABCDE, angle A = 110 degrees, angle B = 140 degrees and angle D = angle E. The sides AB and DC, when produced, meet at right angle. Calculate angles BCD and E.

Solution:

According to the given statement, the figure will be as shown below.

In the figure, AB and DC produced meet at point P, therefore ∠P = 90 degrees

∠B = 140 degrees, which gives ∠PBC = 180 degrees - 140 degrees = 40 degrees

and ∠BCP = 90 degrees - 40 degrees = 50 degrees

∠BCD = 180 degrees - 50 degrees = 130 degrees

Let angle D = angle E = x

Since, ∠A + ∠B + ∠BCD + ∠D + ∠E = (2 times 5 - 4) times 90 degrees

which gives 110 degrees + 140 degrees + 130 degrees + x + x = 540 degrees

i.e. 2x = 540 degrees - 380 degrees = 160 degrees, which gives x = 80 degrees

Angle E = x = 80 degrees

Solved Example 5

By dividing into triangles, find the sum of the angles of the doubly re-entrant heptagon ABCDEFG as shown alongside. Does the general value of (2n - 4) right-angles hold for re-entrant polygon?

Solution:

On dividing the given figure into triangles, we get 5 triangles.

Sum of the angles of given heptagon = Sum of the angles of 5 triangles = 5 times 180 degrees = 900 degrees

n = 7

(2n - 4) right angles = (2 times 7 - 4) times 90 degrees = 10 times 90 degrees = 900 degrees

General value (2n - 4) right angles holds for re-entrant polygons.

Regular Polygon

If all the sides and all the angles of a polygon are equal, it is called a regular polygon.

A regular polygon means a polygon with:

all its sides equal to each other,

all its interior angles equal to each other

and, all its exterior angles also equal to each other.

Sum of interior angles of an 'n' sided polygon (whether it is regular or not) = (2n - 4) rt. angles

and sum of its exterior angles = 4 right angles = 360 degrees.

At each vertex of every polygon, Exterior angle + Interior angle = 180 degrees.

Each interior angle of a regular polygon = \[\frac{(2n - 4) \text{ rt. angles}}{n} = \frac{(2n - 4) \times 90°}{n}\]

Each exterior angle of a regular polygon = \[\frac{4\text{ rt. angles}}{n} = \frac{360°}{n}\]

Teacher's Note

Understanding polygon angles helps when designing floor tiles or building structures - architects use these angle relationships to ensure walls and corners align perfectly.

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