Class 8 Maths Chapter 4: Quadrilaterals Rectangles, Squares, Parallelograms & Rhombus Explained

Class 8 Maths – Chapter 4: Quadrilaterals
(Ganita Prakash Part 1)

Introduction

A quadrilateral is a four‑sided closed figure. This chapter explores different types of quadrilaterals such as rectangles, squares, parallelograms, and rhombuses. Students learn how to deduce geometric properties using reasoning, congruence, and construction.

Basic Knowledge Required

• Types of angles

• Triangle congruence (SAS, AAS, SSS)

• Parallel lines and transversal properties

• Construction using ruler, compass, set‑square

• Understanding of diagonals

Important Definitions

Quadrilateral

A closed figure with four sides.

Rectangle

A quadrilateral with:

1. All angles = 90°

2. Opposite sides equal

OR

A quadrilateral whose diagonals are equal and bisect each other.

Square

A quadrilateral with:

1. All angles = 90°

2. All sides equal

3. Diagonals equal and bisect at 90°

Parallelogram

A quadrilateral with both pairs of opposite sides parallel.

Rhombus

A quadrilateral with all sides equal.

Formulas / Concepts Used

• Sum of angles of a quadrilateral = 360°

• Opposite sides of parallelogram are equal

• Adjacent angles in parallelogram sum to 180°

• Diagonals of rectangle: equal & bisect

• Diagonals of square: equal, bisect at 90°

• Diagonals of parallelogram: bisect

• Diagonals of rhombus: bisect at 90°

Solved Examples (Step‑by‑Step)

Example 1: Carpenter’s Problem (Rectangle Construction)

Given one diagonal = 8 cm.

To form a rectangle:

• Both diagonals must be equal → other diagonal = 8 cm

• Diagonals must bisect each other → join at midpoint

• Angle between diagonals can be any value → still rectangle

Example 2: Angle Deduction in Rectangle

Using congruent triangles, diagonals bisect each other.

Thus OA = OC and OB = OD.

Example 3: Square Diagonal Angle

In a square, diagonals bisect at 90°.

Reason:

Triangles formed by diagonals are congruent (SSS).

Thus ∠BOA = ∠BOC = 90°.

Example 4: Rhombus Angle Calculation

Given one angle = 50°.

Diagonal divides angle into two equal parts.

Let each part = a.

a + a + 50 = 180 → a = 65°

Thus angles = 50°, 130°, 50°, 130°.

 FIGURE IT OUT — COMPLETE SOLUTIONS

SECTION 4.1 — Rectangles and Squares

1. Find all angles in the rectangles

(i) Using right angles and congruent triangles, angles alternate between 30°, 60°.

(ii) Same reasoning: angles alternate between given values.

2. Draw quadrilaterals with diagonals 8 cm, bisecting each other at angles:

(i) 30° → rectangle

(ii) 40° → rectangle

(iii) 90° → square

(iv) 140° → rectangle

3. APML with perpendicular diameters

Two perpendicular diameters form a square.

4. Making 90° using two sticks & thread

• Fix stick A

• Tie thread to both sticks

• Pull thread tight to form equal diagonals

• When diagonals bisect at midpoint → angle = 90°

5. Are opposite sides parallel enough to define a rectangle?

No.

A parallelogram also has opposite sides parallel and equal, but angles are not 90°.

Thus this definition is incomplete.

SECTION 4.2 — Angles in a Quadrilateral

Sum of angles

Using diagonal division:

Sum = 180° + 180° = 360°.

Thus impossible to have three right angles and one non‑right angle.

SECTION 4.3 — Parallelograms

1. Remaining angles in parallelogram

Given angle = 30°

Opposite angle = 30°

Adjacent angles = 150°

2. Opposite angles equal

Using transversal properties:

If one angle = x, opposite = x.

3. Opposite sides equal

Using congruent triangles ABD and CDB → AB = CD, AD = BC.

4. Diagonals bisect

Using triangles AOE and COY → diagonals bisect each other.

SECTION 4.4 — Rhombus

1. Remaining angles

Given angle = 50°

Opposite = 50°

Other two = 130°

2. Rhombus is a parallelogram

Opposite sides parallel → parallelogram

All sides equal → rhombus

Square belongs to both sets.

 Conclusion

Quadrilaterals form a rich family of geometric shapes. Understanding their properties through reasoning, congruence, and construction helps build strong geometric intuition.

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FAQs – Chapter 4: Quadrilaterals

1. What is a quadrilateral?

A quadrilateral is a closed figure with four sides.

2. What defines a rectangle?

A rectangle has all angles 90° and opposite sides equal, or equivalently, diagonals equal and bisecting each other.

3. What defines a square?

A square has all sides equal and all angles 90°, with diagonals equal and perpendicular.

4. Do diagonals of a rectangle bisect each other?

Yes, they bisect each other and are equal in length.

5. What is the sum of angles in a quadrilateral?

The sum of all interior angles is always 360°.

6. What is a parallelogram?

A quadrilateral with both pairs of opposite sides parallel.

7. Are opposite angles equal in a parallelogram?

Yes, opposite angles are always equal.

8. Do diagonals of a parallelogram bisect each other?

Yes, they bisect each other but are not necessarily equal.

9. What is a rhombus?

A quadrilateral with all sides equal.

10. Are diagonals of a rhombus perpendicular?

Yes, they bisect each other at 90°.

11. Is every square a rectangle?

Yes, because it has all angles 90° and opposite sides parallel.

12. Is every rectangle a parallelogram?

Yes, because opposite sides are parallel.

13. Can a quadrilateral have three right angles?

No, because the fourth must also be 90° to make the sum 360°.

14. What angle do diagonals of a square make?

They intersect at 90°.

15. How to identify a parallelogram?

Opposite sides equal, opposite sides parallel, opposite angles equal, diagonals bisect each other.