CBSE Class 8 Maths Revision Notes Chapter 4: Quadrilaterals

Quadrilaterals are closed figures made using four sides, four vertices and four angles. Their sides, angles and diagonals help us identify rectangles, squares, parallelograms, rhombuses, kites and trapeziums.

A quadrilateral is a closed shape formed by four line segments. Different quadrilaterals are classified according to their side lengths, angles, parallel sides and diagonal properties.

These CBSE Class 8 Maths Revision Notes Chapter 4 explain rectangles, squares, parallelograms, rhombuses, kites, trapeziums and isosceles trapeziums. They also cover the angle sum of a quadrilateral, relationships among different quadrilaterals and the use of diagonals for identifying and constructing shapes.

Key Takeaways

• Angle sum: The four interior angles of every quadrilateral add up to 360°.
• Rectangle: Its four angles are 90°, and its diagonals are equal and bisect each other.
• Parallelogram: Its opposite sides are parallel and equal, while its diagonals bisect each other.
• Rhombus: All four sides are equal, and its diagonals bisect each other at 90°.

Access Class 8 Maths Chapter 4 Quadrilaterals Notes in 30 Minutes

Spend the first 10 minutes revising rectangles, squares and the angle sum property. Use the next 10 minutes for parallelograms and rhombuses.

Use the final 10 minutes for kites, trapeziums, diagonal properties and the relationship among different quadrilaterals.

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What Is a Quadrilateral?

A quadrilateral is a closed figure with four sides, four vertices and four angles. The word quadrilateral comes from the Latin words quadri, meaning four, and latus, meaning side.

The four sides must join at their endpoints to form a closed figure. Rectangles, squares, parallelograms, rhombuses, kites and trapeziums are all quadrilaterals.

Main Parts of a Quadrilateral

A quadrilateral has:

• Four sides
• Four vertices
• Four interior angles
• Two diagonals

A diagonal is a line segment joining two non-adjacent vertices.

For example, in quadrilateral ABCD:

• AB, BC, CD and DA are sides.
• A, B, C and D are vertices.
• AC and BD are diagonals.

Rectangle in Class 8 Maths Chapter 4 Notes

A rectangle is a quadrilateral in which all four angles are equal to 90°.

The opposite sides of a rectangle are equal and parallel. Its diagonals are also equal and bisect each other.

Properties of a Rectangle

1. All four angles are 90°.
2. Opposite sides are equal.
3. Opposite sides are parallel.
4. Diagonals are equal.
5. Diagonals bisect each other.

If ABCD is a rectangle and diagonals AC and BD intersect at O, then:

AB = CD

BC = AD

AC = BD

AO = OC

BO = OD

Why Are the Diagonals of a Rectangle Equal?

Consider rectangle ABCD.

In triangles ADC and DAB:

AD = AD, which is a common side.

CD = AB, because opposite sides of a rectangle are equal.

∠ADC = ∠DAB = 90°.

Therefore:

∆ADC ≅ ∆DAB by SAS congruence.

So:

AC = BD

Hence, the diagonals of a rectangle are equal.

Why Do the Diagonals Bisect Each Other?

If diagonals AC and BD meet at O, then:

AO = OC

BO = OD

This means O is the midpoint of both diagonals. Therefore, the diagonals bisect each other.

Diagonal Definition of a Rectangle

A quadrilateral whose diagonals are equal and bisect each other is a rectangle.

This gives another way to identify or construct a rectangle.

Constructing a Rectangle Using Its Diagonals

Suppose one diagonal of a rectangle is 8 cm long. The other diagonal must also be 8 cm because the diagonals of a rectangle are equal.

Both diagonals must be joined at their midpoints. The angle between them may vary, but the resulting quadrilateral will still be a rectangle if the diagonals remain equal and bisect each other.

Construction idea:

1. Draw AC = 8 cm.
2. Mark its midpoint O.
3. Draw another line through O.
4. Mark B and D so that BO = OD = 4 cm.
5. Join A, B, C and D in order.

The quadrilateral formed is a rectangle.

Square in Quadrilaterals Class 8 Notes

A square is a quadrilateral in which all four sides are equal and all four angles are 90°.

A square is a special type of rectangle. Therefore, every square is a rectangle, but every rectangle is not a square.

Properties of a Square

1. All four sides are equal.
2. All four angles are 90°.
3. Opposite sides are parallel.
4. Diagonals are equal.
5. Diagonals bisect each other.
6. Diagonals intersect at 90°.
7. Diagonals bisect the angles of the square.

If ABCD is a square and diagonals AC and BD meet at O, then:

AB = BC = CD = DA

AC = BD

AO = OC

BO = OD

∠AOB = 90°

Each diagonal divides a 90° corner angle into two 45° angles.

Constructing a Square from Its Diagonal

Suppose the diagonal of a square is 8 cm.

1. Draw AC = 8 cm.
2. Mark its midpoint O.
3. Draw a perpendicular line through O.
4. Mark B and D so that BO = OD = 4 cm.
5. Join A, B, C and D.

The diagonals are equal, bisect each other and meet at 90°. Therefore, the resulting quadrilateral is a square.

Rectangle and Square Comparison

Angle Sum of a Quadrilateral

The sum of the four interior angles of any quadrilateral is 360°.

To understand this, draw a diagonal in a quadrilateral. It divides the quadrilateral into two triangles.

The angle sum of each triangle is 180°.

Therefore:

Sum of angles of quadrilateral = 180° + 180°

Sum of angles of quadrilateral = 360°

If the angles are ∠A, ∠B, ∠C and ∠D, then:

∠A + ∠B + ∠C + ∠D = 360°

Example 1

Three angles of a quadrilateral are 80°, 95° and 105°. Find the fourth angle.

Fourth angle = 360° − (80° + 95° + 105°)

Fourth angle = 360° − 280°

Fourth angle = 80°

Example 2

A quadrilateral has three right angles. Find the fourth angle.

Fourth angle = 360° − (90° + 90° + 90°)

Fourth angle = 360° − 270°

Fourth angle = 90°

Therefore, a quadrilateral with three right angles must also have a fourth right angle.

Parallelogram in Class 8 Maths Chapter 4 Revision Notes

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

A rectangle, square and rhombus are all special types of parallelograms.

Properties of a Parallelogram

1. Opposite sides are parallel.
2. Opposite sides are equal.
3. Opposite angles are equal.
4. Adjacent angles add up to 180°.
5. Diagonals bisect each other.
6. Diagonals are not necessarily equal.
7. Diagonals are not necessarily perpendicular.

If ABCD is a parallelogram, then:

AB ∥ CD

BC ∥ AD

AB = CD

BC = AD

∠A = ∠C

∠B = ∠D

∠A + ∠B = 180°

∠B + ∠C = 180°

If diagonals AC and BD meet at O:

AO = OC

BO = OD

Angles of a Parallelogram

Suppose one angle of a parallelogram is 30°.

The opposite angle is also 30°.

Each adjacent angle is:

180° − 30° = 150°

Therefore, the four angles are:

30°, 150°, 30° and 150°

In general, if one angle is x:

Opposite angle = x

Each adjacent angle = 180° − x

Example

If ∠A = 70° in parallelogram ABCD:

∠C = 70°

∠B = 180° − 70° = 110°

∠D = 110°

Why Are Opposite Sides of a Parallelogram Equal?

Draw diagonal BD in parallelogram ABCD.

In triangles ABD and CDB, alternate angles are equal because opposite sides are parallel. BD is also common to both triangles.

The triangles are congruent. Therefore, their corresponding sides are equal.

Hence:

AB = CD

AD = BC

Diagonals of a Parallelogram

The diagonals of a parallelogram bisect each other.

If AC and BD intersect at O:

AO = OC

BO = OD

However, the diagonals are not always equal. Equal diagonals are an additional property of a rectangle.

Parallelogram and Rectangle Comparison

Rhombus in Properties of Quadrilaterals

A rhombus is a quadrilateral in which all four sides are equal.

Every rhombus is also a parallelogram because its opposite sides are parallel. However, its angles do not have to be 90°.

Properties of a Rhombus

1. All four sides are equal.
2. Opposite sides are parallel.
3. Opposite angles are equal.
4. Adjacent angles add up to 180°.
5. Diagonals bisect each other.
6. Diagonals intersect at 90°.
7. Diagonals bisect the angles.
8. Diagonals are not necessarily equal.

If ABCD is a rhombus:

AB = BC = CD = DA

AB ∥ CD

BC ∥ AD

∠A = ∠C

∠B = ∠D

Example

If one angle of a rhombus is 50°, the opposite angle is also 50°.

Each adjacent angle is:

180° − 50° = 130°

Therefore, the angles are:

50°, 130°, 50° and 130°

Diagonals of a Rhombus

The diagonals of a rhombus have three important properties:

• They bisect each other.
• They meet at 90°.
• They bisect the opposite angles.

Suppose AC and BD are diagonals meeting at O.

Then:

AO = OC

BO = OD

AC ⟂ BD

The diagonal AC bisects ∠A and ∠C.

The diagonal BD bisects ∠B and ∠D.

Constructing a Rhombus from Its Diagonals

Suppose the diagonals are 5 cm and 4 cm.

1. Draw AC = 5 cm.
2. Mark midpoint O.
3. Draw a perpendicular line through O.
4. Mark B and D so that BO = OD = 2 cm.
5. Join A, B, C and D.

Since the diagonals bisect each other at 90°, the resulting quadrilateral is a rhombus.

Rhombus and Square Comparison

A square is both a rectangle and a rhombus.

Relationship Among Quadrilaterals

Different quadrilaterals are related because some satisfy the definitions of more than one shape.

• Every square is a rectangle.
• Every square is a rhombus.
• Every square is a parallelogram.
• Every rectangle is a parallelogram.
• Every rhombus is a parallelogram.
• Every parallelogram is a quadrilateral.
• Every rectangle is not a square.
• Every rhombus is not a square.
• Every parallelogram is not a rectangle.

This relationship can be visualised as:

Quadrilateral
→ Parallelogram
→ Rectangle
→ Square

Quadrilateral
→ Parallelogram
→ Rhombus
→ Square

Kite and Trapezium

Kites and trapeziums are also important quadrilaterals. Their definitions depend mainly on equal adjacent sides and parallel sides.

Kite

A kite is a quadrilateral having two non-overlapping pairs of equal adjacent sides.

If ABCD is a kite:

AB = BC

CD = DA

The equal sides meet at two opposite vertices.

Properties of a Kite

1. It has two pairs of equal adjacent sides.
2. One diagonal bisects the other.
3. The diagonals intersect at 90°.
4. One diagonal bisects a pair of opposite angles.
5. One pair of opposite angles is equal.

If diagonal BD is the line joining the vertices where equal sides meet, then BD:

• Bisects ∠B
• Bisects ∠D
• Bisects diagonal AC
• Is perpendicular to AC

Kite and Rhombus

A rhombus satisfies the definition of a kite because it has two pairs of equal adjacent sides. Therefore, every rhombus can be considered a kite.

However, every kite is not a rhombus because all four sides of a kite need not be equal.

Trapezium

A trapezium is a quadrilateral with at least one pair of parallel opposite sides.

If PQRS is a trapezium and:

PQ ∥ SR

then the angles on each non-parallel side are supplementary.

∠P + ∠S = 180°

∠Q + ∠R = 180°

Example

If ∠P = 135°:

∠S = 180° − 135°

∠S = 45°

If ∠Q = 105°:

∠R = 180° − 105°

∠R = 75°

Isosceles Trapezium

An isosceles trapezium is a trapezium whose non-parallel sides are equal.

If UVWX is an isosceles trapezium and UV ∥ XW, with UX = VW, then:

∠U = ∠V

∠X = ∠W

The angles adjacent to the same parallel side are equal.

Properties of an Isosceles Trapezium

1. One pair of opposite sides is parallel.
2. Non-parallel sides are equal.
3. Angles at one base are equal.
4. Angles at the other base are equal.

Trapezium and Parallelogram Comparison

Diagonal Properties of Quadrilaterals

The properties of diagonals are useful for identifying shapes.

How to Identify a Quadrilateral from Its Diagonals

Equal Diagonals That Bisect Each Other

The quadrilateral is a rectangle. It will be a square only if the diagonals are also perpendicular.

Diagonals That Bisect Each Other

The quadrilateral is a parallelogram.

Perpendicular Diagonals That Bisect Each Other

The quadrilateral is a rhombus. It becomes a square if the diagonals are also equal.

Equal, Perpendicular Diagonals That Bisect Each Other

The quadrilateral is a square.

Important True and False Concepts

A quadrilateral whose diagonals are equal and bisect each other must be a square.

False. It may be a rectangle whose diagonals are not perpendicular.

A quadrilateral with three right angles must be a rectangle.

True. The fourth angle is also 90° because the total is 360°.

A quadrilateral whose diagonals bisect each other is a parallelogram.

True.

A quadrilateral whose diagonals are perpendicular must be a rhombus.

False. A kite can also have perpendicular diagonals.

A quadrilateral whose opposite angles are equal is a parallelogram.

True.

A quadrilateral in which all angles are equal is a rectangle.

True. Each angle must be 90° because:

360° ÷ 4 = 90°

Every isosceles trapezium is a parallelogram.

False. It may have only one pair of parallel sides.

Class 8 Maths Chapter 4 Summary Notes

Important Terms from Quadrilaterals

Quadrilateral: A closed figure with four sides.

Diagonal: A line segment joining two non-adjacent vertices.

Rectangle: A quadrilateral with four right angles.

Square: A quadrilateral with four equal sides and four right angles.

Parallelogram: A quadrilateral with both pairs of opposite sides parallel.

Rhombus: A quadrilateral with all four sides equal.

Kite: A quadrilateral with two pairs of equal adjacent sides.

Trapezium: A quadrilateral with at least one pair of parallel opposite sides.

Isosceles trapezium: A trapezium whose non-parallel sides are equal.

Bisect: To divide something into two equal parts.

Perpendicular lines: Lines that meet at an angle of 90°.

Parallel lines: Lines that remain the same distance apart and never meet.

Adjacent angles: Angles that share a common side and vertex.

Opposite angles: Angles lying across from each other in a quadrilateral.

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