Quadrilaterals – Class 8 Maths Chapter 4 Notes & NCERT PDF

Class 8MathematicsChapter 4NCERT book: Ganita Prakash - Mathematics Class 8

Notes and a simple summary of Chapter 4 of the NCERT Class 8 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 4: Quadrilaterals

Introduction

A quadrilateral is a closed figure with four sides — from the Latin quadri (four) and latus (side). This chapter doesn't just list properties. It deduces them step by step using congruent triangles, parallel lines and angle sums, and then checks them by construction.

Measuring a few figures only gives a conjecture. A property is certain only when we justify (prove) it — we can't be sure the 1000th rectangle will behave the same unless we reason it out.

Key Ideas

4.1 Rectangles and squares

A Carpenter's Problem. A carpenter has one 8 cm wooden strip and must join a second strip so that a thread through their endpoints makes a rectangle. The strips are the diagonals. Three deductions answer the question:

Deduction 1 — length of the other diagonal: ΔADC ≅ ΔDAB (SAS: AB = CD, ∠A = ∠D = 90°, AD common), so AC = BD. The diagonals of a rectangle are equal → the other strip is also 8 cm.

Deduction 2 — where they meet: ΔAOB ≅ ΔCOD (AAS), so OA = OC and OB = OD. The diagonals bisect each other (meet at their midpoints).

Deduction 3 — the angle between them: it can be any angle. With equal diagonals bisecting each other at angle x, the base angles are 90° − x/2 and x/2, which add up to 90° at every corner.

Rectangle (new definition): a quadrilateral whose diagonals are equal and bisect each other.

Carpenters in Europe and farmers in Mozambique use exactly this method to mark rectangles.

Deduction 4: if all four angles are 90°, the opposite sides must be equal (ΔBAD ≅ ΔDCB by AAS). So the simplest definition is:

Rectangle: a quadrilateral in which all the angles are 90°.

Properties of a rectangle

All angles are 90°.

Opposite sides are equal.

Opposite sides are parallel (co-interior angles 90° + 90° = 180°).

Diagonals are equal and bisect each other.

Square — a special rectangle. A square has all angles 90° and all sides equal. Every square is a rectangle, but not every rectangle is a square (like being both Indian and Malayali). This is shown with a Venn diagram: the "square" region sits inside the "rectangle" region.

Deduction 5: in a square, ΔBOA ≅ ΔBOC (SSS), so ∠BOA = ∠BOC. They add up to 180°, so each is 90°. The diagonals of a square bisect each other at right angles.

A diagonal splits each 90° corner into 45° + 45°, so the diagonals of a square bisect its angles.

4.2 Angles in a quadrilateral

Draw a diagonal of quadrilateral SOME. It splits into two triangles, each with angle sum 180°, so:

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

That is why a quadrilateral can't have exactly three right angles: the fourth would be 360° − 270° = 90° too.

4.3 Parallelograms

Parallelogram: a quadrilateral whose opposite sides are parallel.

A rectangle is a special parallelogram. Construct one with sides 4 cm and 5 cm and an angle of 30°: draw the two sides, then a line through each free end parallel to the other side.

Deduction 6: adjacent angles are co-interior angles, so they add to 180°. Opposite angles are equal (angles x and 180° − x). In the 30° example the angles are 30°, 150°, 30°, 150°.

Deduction 7: ΔABD ≅ ΔCDB (AAS), so opposite sides are equal.

Deduction 8: in parallelogram EASY, ΔAOE ≅ ΔYOS (ASA), so the diagonals bisect each other. They are not necessarily equal.

4.4 Rhombus

Rhombus: a quadrilateral with all four sides equal.

Deduction 9: a diagonal makes two congruent isosceles triangles, so the four angles it forms at the corners are all equal. With one angle 50°, each is (180 − 50) ÷ 2 = 65°, and the rhombus angles are 50°, 130°, 50°, 130°.

The equal alternate angles show the opposite sides are parallel, so every rhombus is a parallelogram. A square is a rectangle and a rhombus.

Deduction 10: ΔGEO ≅ ΔMEO, so the diagonals meet at 90°.

Properties of a rhombus

All sides are equal.

Opposite sides are parallel.

Adjacent angles add to 180°, and opposite angles are equal.

Diagonals bisect each other at right angles.

Diagonals bisect the angles.

4.5 Playing with quadrilaterals

Geoboard: equal, perpendicular diagonals that bisect each other give a square. Extend one diagonal by 2 cm on both sides and you get a rhombus.

Joining triangles: two equilateral triangles of side 8 cm make a rhombus. Two isosceles triangles (8, 8, 6 cm) and two scalene triangles (6, 9, 12 cm) can be joined in different ways to make parallelograms, kites and other shapes.

4.6 Kite and trapezium

Kite: a quadrilateral ABCD with AB = BC and CD = DA (two pairs of equal adjacent sides).

In a kite, the diagonal BD bisects ∠ABC and ∠ADC, and bisects AC at right angles (hint: ΔAOB ≅ ΔCOB).

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

If PQ ∥ SR, then ∠S + ∠P = 180° and ∠R + ∠Q = 180°. In an isosceles trapezium, the non-parallel sides are equal, and the angles opposite the equal sides are equal (∠U = ∠V, proved by drawing perpendiculars to form a rectangle and congruent triangles).

Worked Examples

Rectangle diagonals meeting at 60°. ΔAOB is isosceles (OA = OB), so a + a + 60 = 180 → a = 60°. At the 120° angle, the base angles are (180 − 120) ÷ 2 = 30°. Each corner is 60° + 30° = 90° ✓.

Angles of a parallelogram with one angle 30°. Adjacent angle = 180 − 30 = 150°; opposite angles are equal → 30°, 150°, 30°, 150°.

Rhombus with one angle 50°. The diagonal splits it into isosceles triangles: a = (180 − 50) ÷ 2 = 65°. Angles are 50°, 130°, 50°, 130°.

Trapezium PQRS with PQ ∥ SR, ∠Q = 105° and ∠P = 135°. ∠R = 180 − 105 = 75° and ∠S = 180 − 135 = 45°.

Isosceles trapezium with ∠D = 100° (AD = BC). ∠A = 180 − 100 = 80°, and ∠B = ∠A = 80°.

Constructing a square with a 6 cm diagonal without a protractor. Draw AB = 6 cm, draw its perpendicular bisector with a compass, mark C and D on it 3 cm from the midpoint O, then join A, C, B, D.

Figure it Out Hints

To find angles inside a rectangle, use: diagonals equal and bisecting each other → isosceles triangles; vertically opposite angles are equal; angles on a line add to 180°.

To construct a quadrilateral from its diagonals, draw one diagonal, mark its midpoint, draw the given angle there, and mark half the other diagonal on each side.

Two perpendicular diameters of a circle give a square (equal diagonals bisecting at 90°).

For true/false statements, look for a counter-example. Equal bisecting diagonals give a rectangle, not necessarily a square. Perpendicular diagonals could give a kite, not just a rhombus.

If opposite sides are equal, draw a diagonal: SSS congruence gives equal alternate angles, so the shape is a parallelogram.

Maths Words

Quadrilateral — चतुर्भुज

Diagonal — विकर्ण

Rectangle — आयत

Square — वर्ग

Parallelogram — समांतर चतुर्भुज

Rhombus — समचतुर्भुज

Kite — पतंग

Trapezium — समलंब चतुर्भुज

Bisect — समद्विभाजित करना

Congruent — सर्वांगसम

Transversal — तिर्यक रेखा

Conjecture — अनुमान

Venn diagram — वेन आरेख

Perpendicular — लंबवत

Common Mistakes

Thinking the diagonals of every parallelogram are equal — only in rectangles (and squares).

Saying "a square is not a rectangle" — a square is a special rectangle and a special rhombus.

Believing perpendicular diagonals always mean a rhombus — a kite also has perpendicular diagonals.

Writing congruences in the wrong order (e.g. ΔBAD ≅ ΔCDB instead of ΔDCB) — the vertices must correspond.

Treating measurement as proof — measuring gives a conjecture, and deduction proves it.

Maths in Daily Life

Carpenters check a door frame is rectangular by measuring whether its diagonals are equal.

Kites, floor tiles, window grills and table tops show rhombuses, trapeziums and parallelograms.

Farmers in Mozambique lay out rectangular house bases using two equal ropes crossing at their midpoints.

💡 Key Learning Points

✓Properties of quadrilaterals are proved by deduction (mostly congruent triangles); measurement only gives a conjecture.
✓A rectangle is a quadrilateral with all angles 90°; equivalently, one whose diagonals are equal and bisect each other.
✓The angle sum of any quadrilateral is 360° (split it into two triangles by a diagonal).
✓Parallelograms have opposite sides parallel and equal, opposite angles equal, adjacent angles supplementary and diagonals that bisect each other; rhombuses add equal sides and perpendicular, angle-bisecting diagonals.
✓Squares, rectangles, rhombuses and parallelograms nest inside each other (Venn diagram); kites have two pairs of equal adjacent sides, and trapeziums have at least one pair of parallel sides.

👨‍🏫 Teaching Tips

→Act out the Carpenter's Problem with two equal sticks and a thread: let students discover that crossing them at their midpoints always gives a rectangle.
→For every property, first let students measure and form a conjecture, then guide them to the congruence proof — stress the difference.
→Use a geoboard or dot paper to build squares, rhombuses and kites from diagonals; extend one diagonal to turn a square into a rhombus.
→Build the quadrilateral Venn diagram as a class poster, placing squares inside both rectangles and rhombuses.
→Cut out pairs of equilateral, isosceles (8, 8, 6 cm) and scalene (6, 9, 12 cm) triangles and let groups find every quadrilateral they can make.
→Practise writing congruences with vertices in the correct order, using the book's 'Is it wrong to write ΔBAD ≅ ΔCDB?' prompts.
→Use the 'Which Quad?' paper-folding game at the end of the chapter to connect creases with diagonals.

📋 Assessment Questions

Can the student prove that the diagonals of a rectangle are equal and bisect each other?
Can the student find all remaining angles of a parallelogram, rhombus or trapezium given one or two angles?
Can the student construct a square, rhombus, parallelogram or kite from given diagonals?
Can the student place square, rectangle, rhombus, parallelogram and kite correctly in a Venn diagram and justify it?
Can the student decide whether statements like 'perpendicular diagonals means rhombus' are true, with a counter-example if false?

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