Quadrilaterals — Class 8

What Quadrilaterals Are All About

A quadrilateral is any closed shape with four straight sides. That simple definition opens up a whole family — squares, rectangles, parallelograms, rhombuses, trapeziums, and kites — each with its own properties and tricks.

For Class 8 students, this chapter is the first deep dive into geometry beyond triangles. Every figure here builds intuition for Class 9-10 geometry, and concepts (like properties of diagonals) come back in JEE-level coordinate geometry.

We will work through each type, see how they relate to each other, and lock in the properties you’ll need for board exams.

Key Terms & Definitions

Quadrilateral: A four-sided polygon. The interior angles always sum to $360°$ .

Sides: The four straight edges (denoted $AB, BC, CD, DA$ for quadrilateral $ABCD$ ).

Diagonals: Line segments connecting opposite corners ( $AC$ and $BD$ ).

Convex quadrilateral: All interior angles less than $180°$ — the diagonals lie inside the figure.

Concave quadrilateral: One interior angle greater than $180°$ — one diagonal goes outside.

The Quadrilateral Family Tree

Let’s organize the family by properties:

Trapezium

A trapezium has at least one pair of parallel sides. The parallel sides are called the bases, and the non-parallel sides are the legs.

In some definitions (especially in India and Europe), a trapezium has EXACTLY one pair of parallel sides. In American math, “trapezoid” means at least one pair.

Parallelogram

A parallelogram has TWO pairs of parallel sides. Properties:

Opposite sides are equal in length.
Opposite angles are equal.
Diagonals bisect each other (cut each other in half).
Adjacent angles are supplementary (sum to $180°$ ).

Rectangle

A rectangle is a parallelogram with all angles equal to $90°$ . So all parallelogram properties apply, plus:

All four angles are right angles.
Diagonals are equal in length.

Rhombus

A rhombus is a parallelogram with all four sides equal. Properties:

All sides equal.
Diagonals bisect each other AT RIGHT ANGLES.
Diagonals bisect the angles of the rhombus.

Square

A square is a rhombus with right angles, OR a rectangle with all sides equal. It has every property of rectangles AND rhombuses:

All sides equal.
All angles $90°$ .
Diagonals equal in length, perpendicular, and bisect each other.

Kite

A kite has two pairs of adjacent (consecutive) sides equal. Properties:

One pair of opposite angles equal (the angles between unequal sides).
Diagonals perpendicular.
One diagonal bisects the other.

Methods/Concepts

Property: Angle Sum

Every quadrilateral’s interior angles sum to $360°$ . This follows from dividing the quadrilateral into two triangles (each with $180°$ ).

$\angle A + \angle B + \angle C + \angle D = 360°$

Property: Parallelogram Diagonals

In a parallelogram, the diagonals bisect each other. If $O$ is the intersection point: $AO = OC$ and $BO = OD$ .

This is a useful tool — given any parallelogram problem, drawing the diagonals immediately creates pairs of equal segments.

Method: Identifying Special Quadrilaterals

Given coordinates of four points, identify which quadrilateral they form:

Compute side lengths (use distance formula).
Compute slopes of sides (find parallel pairs).
Compute diagonal lengths (test for rectangle/square/rhombus).

Solved Examples

Easy — Find the Missing Angle

Three angles of a quadrilateral are $80°, 95°$ , and $110°$ . Find the fourth angle.

Sum is $360°$ . So fourth angle $= 360 - 80 - 95 - 110 = 75°$ .

Medium — Identify the Quadrilateral

A parallelogram has diagonals of length $10 \, \text{cm}$ each, intersecting at right angles. What is it?

Equal diagonals → rectangle property. Perpendicular diagonals → rhombus property. Both → square.

Hard — Apply Multiple Properties

In a parallelogram $ABCD$ , $\angle A = 70°$ . Find $\angle B, \angle C, \angle D$ .

Adjacent angles are supplementary: $\angle B = 180 - 70 = 110°$ . Opposite angles are equal: $\angle C = \angle A = 70°$ and $\angle D = \angle B = 110°$ .

Exam-Specific Tips

CBSE Class 8

Boards often ask for “name the quadrilateral with these properties” — practice the family tree well. Also expect 1-2 mark questions on angle sum.

For 5-mark questions, they may give coordinates and ask which special quadrilateral they form.

Beyond Class 8

This foundation supports coordinate geometry in Class 9-10 (proving figures using slope and distance) and conic sections in JEE (where rectangles, ellipses, and circles share properties).

Common Mistakes to Avoid

Mistake 1: Thinking every parallelogram is a rectangle. Only parallelograms with right angles are rectangles.

Mistake 2: Saying a square is “not a rectangle.” A square IS a special type of rectangle (with all sides equal). Every square is also a rhombus.

Mistake 3: Forgetting that the angle sum is $360°$ , not $180°$ . A common slip after working with triangles.

Mistake 4: Assuming kites have all sides equal. Only adjacent pairs are equal — not all four.

Mistake 5: Confusing “diagonals bisect each other” with “diagonals are equal.” The first is true for any parallelogram. The second is only true for rectangles and squares.

Practice Questions

Q1. In a parallelogram $PQRS$ , $\angle P = 75°$ . Find all other angles.

$\angle Q = 105°$ , $\angle R = 75°$ , $\angle S = 105°$ .

Q2. A rhombus has diagonals $6 \, \text{cm}$ and $8 \, \text{cm}$ . Find its side length.

Half-diagonals are $3$ and $4$ . By Pythagoras (since diagonals are perpendicular), side $= \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{cm}$ .

Q3. Is a square a special type of trapezium?

Yes (in the inclusive definition) — a square has parallel sides, satisfying the trapezium definition.

Q4. Three angles of a quadrilateral are equal. The fourth is $108°$ . Find each of the equal angles.

$3x + 108 = 360$ , so $x = 84°$ . Each equal angle is $84°$ .

Q5. A kite has angles $A = 100°$ , $C = 80°$ . The other two angles ( $B$ and $D$ ) are equal. Find $B$ .

$100 + 80 + 2B = 360$ , so $B = 90°$ .

Q6. In a parallelogram, one angle is $30°$ more than the adjacent angle. Find both.

Adjacent angles in a parallelogram are supplementary. Let one angle be $x$ . Then $x + (x + 30) = 180$ , so $x = 75°$ and the other is $105°$ .

Q7. What is the difference between a square and a rhombus?

A rhombus has all sides equal but angles need not be $90°$ . A square has all sides equal AND all angles $90°$ . So every square is a rhombus, but not every rhombus is a square.

Q8. The angles of a quadrilateral are in the ratio $3:4:5:6$ . Find each angle.

Let angles be $3x, 4x, 5x, 6x$ . Sum $= 18x = 360°$ , so $x = 20°$ . Angles: $60°, 80°, 100°, 120°$ .

FAQs

Why does the angle sum equal $360°$ ? A quadrilateral can be split into two triangles by a diagonal. Each triangle’s angles sum to $180°$ , so the total is $360°$ .

Is a parallelogram a trapezium? In the inclusive definition (yes — at least one pair parallel), every parallelogram is also a trapezium. In the exclusive definition (exactly one pair parallel), parallelograms are NOT trapeziums.

What’s the easiest way to identify a rhombus from coordinates? Check that all four sides are equal in length using the distance formula. If yes, it’s a rhombus.

Are all rectangles squares? No. Squares are rectangles with all sides equal. A typical rectangle has sides of two different lengths.

Can a quadrilateral have three obtuse angles? Yes. As long as the fourth angle is small enough to keep the sum at $360°$ . Example: $100°, 110°, 120°, 30°$ .

What is the only quadrilateral whose diagonals are perpendicular AND equal? A square. Rhombus has perpendicular diagonals but unequal lengths; rectangle has equal but not perpendicular.

Why do diagonals of a rhombus bisect angles? Because a rhombus has all sides equal, the triangles formed by the diagonals are isosceles, so the diagonals split the angles symmetrically.

Is a kite a parallelogram? No. A parallelogram has opposite sides equal; a kite has adjacent sides equal. They are different.