ICSE Class 8 Maths Chapter 27 Quadrilateral

Chapter 27: Quadrilateral

27.1 Review

Quadrilateral: A quadrilateral is a closed polygon with four sides.

The adjoining figure shows a quadrilateral ABCD which has:

(i) Four sides: AB, BC, CD and DA

(ii) Four vertices: A, B, C and D

(iii) Four angles: \(\angle ABC\), \(\angle BCD\), \(\angle CDA\) and \(\angle DAB\)

(iv) Two diagonals: AC and BD

The sum of angles of a quadrilateral = 4 right angles = 360°.

Example 1

The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Find all its angles.

Solution

Since, 3 + 4 + 5 + 6 = 18 and sum of the angles of a quadrilateral is 360°.

First angle = \(\frac{3}{18} \times 360° = 60°\), second angle = \(\frac{4}{18} \times 360° = 80°\),

third angle = \(\frac{5}{18} \times 360° = 100°\) and, fourth angle = \(\frac{6}{18} \times 360° = 120°\) (Ans.)

Alternative method

Let the angles of the quadrilateral be 3x, 4x, 5x and 6x.

\(\therefore 3x + 4x + 5x + 6x = 360°\) \(\Rightarrow 18x = 360°\) and \(x = 20°\)

\(\therefore\) First angle = 3x = 3 × 20° = 60°, second angle = 4x = 4 × 20° = 80°,

third angle = 5x = 5 × 20° = 100° and fourth angle = 6x = 6 × 20° = 120° (Ans.)

Example 2

Three angles of a quadrilateral are in the ratio 4 : 6 : 3. If the fourth angle is 100°; find the other three angles of the quadrilateral.

Solution

Let the three angles be 4x, 6x and 3x

\(\therefore 4x + 6x + 3x + 100° = 360°\)

\(\Rightarrow 13x = 360° - 100° = 260°\) and, \(x = \frac{260°}{13} = 20°\)

\(\therefore\) The other three angles are: 4x, 6x and 3x

= 4 × 20°, 6 × 20° and 3 × 20° = 80°, 120° and 60° (Ans.)

Teacher's Note

Understanding angle sums in quadrilaterals helps us determine missing angles in real-world shapes like tiles, window panes, and building layouts.

Test Yourself

1. In quadrilateral ABCD, \(\angle A : \angle B : \angle C : \angle D = 6 : 4 : 5 : 3\). \(\angle A = ................... = ...........\) and \(\angle D = ................... = ........... .\) The special name of this quadrilateral is ....................

2. In quadrilateral ABCD, \(\angle A = 100°\), \(\angle B = 70°\) and \(\angle C : \angle D = 8 : 11\); the angle \(\angle D = ............................................\)

3. If one exterior angle of a quadrilateral is 140° and the other three angles of the quadrilateral are in the ratio 8 : 15 : 9, the largest angle of the quadrilateral ......................................\)

4. Two angles of a quadrilateral are equal and the two other angles are separately equal. Can any two sides of the given quadrilateral be parallel? ..............

Exercise 27 (A)

1. Two angles of a quadrilateral are 89° and 113°. If the other two angles are equal, find the equal angles.

2. Two angles of a quadrilateral are 68° and 76°. If the other two angles are in the ratio 5 : 7, find the measure of each of them.

3. Angles of a quadrilateral are (4x)°, 5(x + 2)°, (7x - 20)° and 6(x + 3)°. Find:

(i) the value of x.

(ii) each angle of the quadrilateral.

4. Use the information given in the following figure to find:

(i) x.

(ii) \(\angle B\) and \(\angle C\).

5. In quadrilateral ABCD, side AB is parallel to side DC. If \(\angle A : \angle D = 1 : 2\) and \(\angle C : \angle B = 4 : 5\).

(i) Calculate each angle of the quadrilateral.

(ii) Assign a special name to quadrilateral ABCD.

6. From the following figure find,

(i) x

(ii) \(\angle ABC\)

(iii) \(\angle ACD\)

7. Given: In quadrilateral ABCD, \(\angle C = 64°\), \(\angle D = \angle C - 8°\); \(\angle A = 5(a + 2)°\) and \(\angle B = 2(2a + 7)°\). Calculate \(\angle A\).

8. In the given figure:

\(\angle b = 2a + 15°\) and \(\angle c = 3a + 5°\), find the values of b and c.

9. Three angles of a quadrilateral are equal. If the fourth angle is 69°, find the measure of equal angles.

10. In quadrilateral PQRS, \(\angle P : \angle Q : \angle R : \angle S = 3 : 4 : 6 : 7\). Calculate each angle of the quadrilateral and then prove that PQ and SR are parallel to each other.

(i) Is PS also parallel to QR?

(ii) Assign a special name to quadrilateral PQRS.

11. Use the information given in the following figure to find the value of x.

12. The following figure shows a quadrilateral in which sides AB and DC are parallel.

If \(\angle A : \angle D = 4 : 5\), \(\angle B = (3x - 15)°\) and \(\angle C = (4x + 20)°\), find each angle of the quadrilateral ABCD.

Teacher's Note

These exercises build problem-solving skills by requiring us to use angle sum properties, which is essential for understanding how different shapes fit together in architectural designs and interior layouts.

27.2 Special Types of Quadrilaterals

1. Trapezium

A trapezium is a quadrilateral in which one pair of opposite sides are parallel but other two sides of it are non-parallel.

If the non-parallel sides of a trapezium are equal, it is called an isosceles trapezium.

In an isosceles trapezium ABCD, side AB // side DC and non-parallel sides AD and BC are equal. Also,

(i) \(\angle A = \angle B\) and \(\angle C = \angle D\)

and, (ii) diagonal AC = diagonal BD.

2. Parallelogram

A parallelogram is a quadrilateral in which both the pairs of opposite sides are parallel.

The given figure shows a parallelogram ABCD as AB is parallel to DC and AD is parallel to BC.

Theorem 5

If a pair of opposite sides of a quadrilateral are equal and parallel, it is a parallelogram.

Given: A quadrilateral ABCD in which AB = DC and AB is parallel to DC.

To prove: ABCD is a parallelogram.

Construction: Join B and D.

Proof

Since, both the pairs of opposite sides of the quadrilateral ABCD are parallel, it is a parallelogram.

Hence Proved.

Teacher's Note

Parallelograms are fundamental shapes used in engineering and design, helping us understand how forces balance and how structures maintain stability in buildings and bridges.

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