Class 8 : Maths – Lesson 4. Quadrilaterals

EXPLANATION AND ANALYSIS

🌍 INTRODUCTION — SHAPES WITH FOUR SIDES

🏙️ When we look around, we notice many shapes that have four sides.
🪟 Windows, 🪑 tabletops, 📺 screens, 🧱 bricks, 🛣️ fields, and 🖼️ picture frames often have four straight boundaries.

📘 Such shapes are called quadrilaterals.

🧠 This lesson explains:

what a quadrilateral is

different types of quadrilaterals

their properties

special quadrilaterals like trapezium, parallelogram, rectangle, square, and rhombus

Understanding quadrilaterals builds a strong base for geometry and helps in recognising shapes correctly.

🔷 WHAT IS A QUADRILATERAL?

📐 A quadrilateral is a closed figure made by four line segments.

✨ Main points:

it has four sides

it has four vertices (corners)

it has four angles

🧩 The word quadrilateral comes from:

quadri → four

lateral → sides

📌 So, quadrilateral means a figure with four sides.

📍 PARTS OF A QUADRILATERAL

🔹 Sides
The four line segments forming the shape.

🔹 Vertices
The four corner points where sides meet.

🔹 Angles
The angles formed at each vertex.

🔹 Diagonals
Line segments joining opposite vertices.

📘 Every quadrilateral has two diagonals.

📐 SUM OF ANGLES OF A QUADRILATERAL

🧠 One important property of quadrilaterals is related to angles.

📌 Angle sum property

The sum of all interior angles of a quadrilateral is 360°.

✨ Explanation idea:
A quadrilateral can be divided into two triangles, and each triangle has angle sum 180°.

So,
180° + 180° = 360°

📐 This property is true for all quadrilaterals.

🧭 TYPES OF QUADRILATERALS

📘 Quadrilaterals are classified based on:

length of sides

measure of angles

parallel sides

Let us study the main types one by one.

🟫 TRAPEZIUM

📐 A trapezium is a quadrilateral in which:

only one pair of opposite sides is parallel

📌 Key features:

one pair of parallel sides

the other pair is not parallel

🧩 Trapezium is often used in bridges, roofs, and road designs.

🟪 PARALLELOGRAM

📐 A parallelogram is a quadrilateral in which:

both pairs of opposite sides are parallel

🔍 Properties of a parallelogram:

opposite sides are equal

opposite angles are equal

diagonals bisect each other

📌 Parallelogram is the base shape for many other special quadrilaterals.

🟥 RECTANGLE

📐 A rectangle is a special parallelogram.

✨ Key properties:

opposite sides are equal and parallel

all angles are right angles (90°)

diagonals are equal

🧠 A rectangle looks like a stretched square.

🏠 Examples:

doors

books

screens

🟩 SQUARE

📐 A square is a special rectangle and also a special parallelogram.

✨ Properties of a square:

all sides are equal

all angles are 90°

diagonals are equal

diagonals bisect each other at right angles

📌 A square has properties of both rectangle and rhombus.

🟦 RHOMBUS

📐 A rhombus is a quadrilateral in which:

all sides are equal

opposite sides are parallel

🔍 Properties:

opposite angles are equal

diagonals bisect each other at right angles

diagonals are not equal

🧩 A rhombus looks like a tilted square.

⚖️ COMPARISON OF QUADRILATERALS

📘 Understanding differences avoids confusion.

🔷 Parallelogram

opposite sides equal

angles not necessarily 90°

🟥 Rectangle

opposite sides equal

all angles 90°

🟩 Square

all sides equal

all angles 90°

🟦 Rhombus

all sides equal

angles not necessarily 90°

📌 Each shape has its own identity and properties.

🧠 DIAGONALS IN QUADRILATERALS

📍 Diagonals behave differently in different quadrilaterals.

✨ Summary of diagonal properties:

Parallelogram → diagonals bisect each other

Rectangle → diagonals are equal

Square → diagonals are equal and perpendicular

Rhombus → diagonals are perpendicular but not equal

📐 Diagonals help in identifying the type of quadrilateral.

🏠 USES OF QUADRILATERALS IN DAILY LIFE

🏗️ Construction of buildings
🪟 Windows and doors
📐 Floor tiles and designs
📊 Graph paper
🧱 Bricks and frames

🌍 Quadrilaterals are widely used in architecture and design.

⚠️ COMMON MISTAKES TO AVOID

🚫 Confusing rectangle with square
🚫 Thinking all parallelograms have right angles
🚫 Forgetting angle sum property
🚫 Mixing properties of rhombus and square

✔️ Always check:

sides

angles

parallelism

diagonals

🌟 IMPORTANCE OF THIS LESSON

🏆 Strengthens geometry foundation
🚀 Helps in understanding polygons
🧠 Improves shape recognition
📘 Useful for higher classes
🌱 Connects maths with real-world shapes

🧾 SUMMARY

📌 A quadrilateral has four sides
📌 Angle sum of a quadrilateral is 360°
📌 Trapezium has one pair of parallel sides
📌 Parallelogram has two pairs of parallel sides
📌 Rectangle has all right angles
📌 Square has equal sides and right angles
📌 Rhombus has equal sides but tilted angles

🔁 QUICK RECAP

🔷 Quadrilateral → four-sided figure
📐 Angle sum → 360°
🟫 Trapezium → one pair parallel
🟥 Rectangle → all angles 90°
🟩 Square → equal sides + right angles
🟦 Rhombus → equal sides, slanted angles

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TEXTBOOK QUESTIONS

🔒 ❓ Q1. Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.
📌 ✅ Answer:
⬥ Each equilateral triangle has all sides 4 cm and each angle 60°.
⬥ When two equilateral triangles of side 4 cm are joined along one side, that common side becomes an internal diagonal, not a side of the quadrilateral.
⬥ The quadrilateral formed has four sides, each of length 4 cm.
⬥ Two angles are formed by joining two 60° angles: 60° + 60° = 120°.
⬥ The remaining two angles are the untouched angles of the equilateral triangles: 60° each.

➡️ Sides: 4 cm, 4 cm, 4 cm, 4 cm
➡️ Angles: 120°, 60°, 120°, 60°

🔒 ❓ Q2. Construct a kite whose diagonals are of lengths 6 cm and 8 cm.
📌 ✅ Answer:
🟢 Step 1 ⬥ Draw a line segment AC = 8 cm (one diagonal).
🔵 Step 2 ⬥ Find the midpoint O of AC and draw a perpendicular line through O.
🟡 Step 3 ⬥ On this perpendicular line, mark points B and D such that OB = OD = 3 cm (half of 6 cm).
🔴 Step 4 ⬥ Join A–B, B–C, C–D, D–A to form the kite.

➡️ The constructed quadrilateral is a kite because one diagonal bisects the other at right angles.

🔒 ❓ Q3. Find the remaining angles in the following trapeziums.

🔒 ❓ (i) First trapezium

📌 ✅ Answer:
⬥ The top and bottom sides are parallel (shown by arrows).
⬥ Consecutive interior angles on the same side of a transversal are supplementary.
⬥ Left bottom angle = 135°, so left top angle = 180° − 135° = 45°.
⬥ Right bottom angle = 105°, so right top angle = 180° − 105° = 75°.

➡️ Remaining angles are 45° and 75°.

🔒 ❓ (ii) Second trapezium

📌 ✅ Answer:
⬥ One pair of opposite sides is parallel (shown by arrows).
⬥ The left pair of non-parallel sides are equal, so it is an isosceles trapezium.
⬥ Base angles of an isosceles trapezium are equal.
⬥ Given one angle = 100°, the adjacent angle on the same base = 100°.
⬥ The other two angles = 180° − 100° = 80° each.

➡️ Remaining angles are 100°, 80°, and 80°.

🔒 ❓ Q4. Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then answer the following questions.

🔒 ❓ (i) What is the quadrilateral that is both a kite and a parallelogram?

📌 ✅ Answer:
⬥ A rhombus has all sides equal (kite property).
⬥ A rhombus also has opposite sides parallel (parallelogram property).

➡️ Answer: Rhombus

🔒 ❓ (ii) Can there be a quadrilateral that is both a kite and a rectangle?

📌 ✅ Answer:
⬥ A kite has two pairs of adjacent equal sides.
⬥ A rectangle has opposite sides equal and all angles 90°.
⬥ These conditions cannot be satisfied together.

➡️ Answer: No

🔒 ❓ (iii) Is every kite a rhombus? If not, what is the correct relationship?

📌 ✅ Answer:
⬥ A kite has two pairs of adjacent equal sides.
⬥ A rhombus has all four sides equal.
⬥ A kite need not have all four sides equal.

➡️ Correct relationship:
Every rhombus is a kite, but every kite is not a rhombus.

🔒 ❓ Q5. If PAIR and RODS are two rectangles, find ∠IOD.
📌 ✅ Answer:
⬥ In rectangle PAIR, angle at R between diagonal RO and base RI is given as 30°.
⬥ Opposite sides of a rectangle are parallel.
⬥ Diagonal RO makes the same angle with the base at O as it does at R.
⬥ Rectangle RODS has right angles at every vertex.
⬥ Angle IOD is formed by extending diagonal RO into the second rectangle.

➡️ ∠IOD = 180° − 30° = 150°

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OTHER IMPORTANT QUESTIONS

🔹 Part A — MCQs (Questions 1–10)

🔒 ❓ Q1. A quadrilateral has diagonals that bisect each other at right angles, but the diagonals are not equal. Which quadrilateral must it be?
🟢1️⃣ Rhombus
🔵2️⃣ Rectangle
🟡3️⃣ Square
🟣4️⃣ Kite
✔️ Answer: 🟢1️⃣

🔒 ❓ Q2. A quadrilateral has exactly one pair of opposite sides parallel and its non-parallel sides are equal. Which statement is always true?
🟢1️⃣ It is a trapezium but not a parallelogram
🔵2️⃣ It is a parallelogram
🟡3️⃣ It is a rhombus
🟣4️⃣ It is a rectangle
✔️ Answer: 🟢1️⃣

🔒 ❓ Q3. If the diagonals of a quadrilateral are equal and bisect each other, which conclusion is logically correct?
🟢1️⃣ It must be a rectangle
🔵2️⃣ It must be a square
🟡3️⃣ It must be a parallelogram
🟣4️⃣ It could be a rectangle or a square
✔️ Answer: 🟣4️⃣

🔒 ❓ Q4. Which condition is sufficient (by itself) to prove a quadrilateral is a parallelogram?
🟢1️⃣ Opposite sides are equal
🔵2️⃣ Diagonals are perpendicular
🟡3️⃣ One angle is 90°
🟣4️⃣ One pair of sides is parallel
✔️ Answer: 🟢1️⃣

🔒 ❓ Q5. A quadrilateral has all sides equal and diagonals that bisect each other at right angles. Which extra condition is needed to make it a square?
🟢1️⃣ One right angle
🔵2️⃣ Diagonals unequal
🟡3️⃣ Adjacent sides unequal
🟣4️⃣ One pair of sides parallel
✔️ Answer: 🟢1️⃣

🔒 ❓ Q6. Which quadrilateral is both a kite and a parallelogram?
🟢1️⃣ Square
🔵2️⃣ Rectangle
🟡3️⃣ Trapezium
🟣4️⃣ General parallelogram
✔️ Answer: 🟢1️⃣

🔒 ❓ Q7. In a quadrilateral, if one diagonal divides it into two congruent triangles, what can be concluded?
🟢1️⃣ The quadrilateral is a parallelogram
🔵2️⃣ The quadrilateral is a square
🟡3️⃣ The quadrilateral is a kite
🟣4️⃣ No definite conclusion
✔️ Answer: 🟢1️⃣

🔒 ❓ Q8. Which statement is never true?
🟢1️⃣ Every square is a rectangle
🔵2️⃣ Every rectangle is a square
🟡3️⃣ Every square is a rhombus
🟣4️⃣ Every square is a parallelogram
✔️ Answer: 🔵2️⃣

🔒 ❓ Q9. If all angles of a quadrilateral are equal, then each angle measures:
🟢1️⃣ 90°
🔵2️⃣ 60°
🟡3️⃣ 120°
🟣4️⃣ 45°
✔️ Answer: 🟢1️⃣

🔒 ❓ Q10. A quadrilateral has diagonals that bisect each other but are not perpendicular. Which figure fits this description?
🟢1️⃣ Parallelogram
🔵2️⃣ Rhombus
🟡3️⃣ Kite
🟣4️⃣ Square
✔️ Answer: 🟢1️⃣

🔹 Part B — Short Answer Questions (Questions 11–20)

🔒 ❓ Q11. Explain why a quadrilateral having three right angles must also have the fourth angle as a right angle.
📌 ✅ Answer:
🔹 The sum of interior angles of a quadrilateral is 360°
🔸 Three right angles sum to 270°, so the fourth must be 90°

🔒 ❓ Q12. Is every kite a rhombus? Justify your answer.
📌 ✅ Answer:
🔹 A kite has two pairs of adjacent equal sides
🔸 A rhombus has all four sides equal, so a kite need not be a rhombus

🔒 ❓ Q13. Why are diagonals of a rectangle equal but not perpendicular in general?
📌 ✅ Answer:
🔹 A rectangle has equal opposite sides and right angles
🔸 Its diagonals bisect each other but meet at right angles only in a square

🔒 ❓ Q14. State one condition that distinguishes a square from a rectangle.
📌 ✅ Answer:
🔹 In a square, all sides are equal
🔸 In a rectangle, only opposite sides are equal

🔒 ❓ Q15. Explain why the diagonals of a parallelogram bisect each other.
📌 ✅ Answer:
🔹 Opposite sides are parallel and equal
🔸 Diagonals form congruent triangles, causing mutual bisection

🔒 ❓ Q16. Can a trapezium have diagonals that bisect each other? Explain.
📌 ✅ Answer:
🔹 Diagonal bisection implies a parallelogram
🔸 A trapezium generally lacks two pairs of parallel sides

🔒 ❓ Q17. Why is every square also a rhombus?
📌 ✅ Answer:
🔹 All sides of a square are equal
🔸 This satisfies the definition of a rhombus

🔒 ❓ Q18. If the diagonals of a quadrilateral bisect the angles, which figure is indicated?
📌 ✅ Answer:
🔹 Angle bisection by diagonals is a property of a rhombus
🔸 A square also satisfies this condition

🔒 ❓ Q19. Why does the sum of interior angles of any quadrilateral equal 360°?
📌 ✅ Answer:
🔹 A quadrilateral can be divided into two triangles
🔸 Each triangle has angle sum 180°, totaling 360°

🔒 ❓ Q20. Distinguish between a kite and a parallelogram using side properties.
📌 ✅ Answer:
🔹 Kite: equal adjacent sides
🔸 Parallelogram: equal opposite sides

🔹 Part C — Detailed Answer Questions (Questions 21–30)

🔒 ❓ Q21. Prove that the diagonals of a rectangle bisect each other but are not perpendicular in general.
📌 ✅ Answer:
🔹 Opposite sides of a rectangle are equal and parallel
🔹 Diagonals divide the rectangle into congruent triangles
🔹 Hence, diagonals bisect each other
🔹 Perpendicularity occurs only if all sides are equal, which is not general

🔒 ❓ Q22. Explain with reasoning whether a quadrilateral with four equal sides and one right angle must be a square.
📌 ✅ Answer:
🔹 Four equal sides imply a rhombus
🔹 One right angle forces all angles to be 90°
🔹 Hence, the quadrilateral is a square

🔒 ❓ Q23. Justify that a quadrilateral whose diagonals bisect each other is a parallelogram.
📌 ✅ Answer:
🔹 Diagonal bisection forms congruent triangles
🔹 Corresponding sides become equal and parallel
🔹 Thus, the quadrilateral is a parallelogram

🔒 ❓ Q24. Explain the relationship between square, rectangle, rhombus, and parallelogram using properties.
📌 ✅ Answer:
🔹 Square satisfies properties of rectangle and rhombus
🔹 Rectangle and rhombus are special parallelograms
🔹 Hence, square ⊂ rectangle ⊂ parallelogram and square ⊂ rhombus

🔒 ❓ Q25. Construct a square of diagonal 6 cm without using a protractor and justify the steps.
📌 ✅ Answer:
🔹 Draw diagonal of length 6 cm
🔹 Construct its perpendicular bisector
🔹 Using equal distances, locate remaining vertices
🔹 All sides become equal with right angles

🔒 ❓ Q26. In a square, midpoints of sides are joined consecutively. Identify the new quadrilateral formed.
📌 ✅ Answer:
🔹 Midpoints are equidistant from center
🔹 Resulting figure has equal sides and right angles
🔹 The new quadrilateral is a square

🔒 ❓ Q27. Prove that the diagonals of a rhombus are perpendicular bisectors of each other.
📌 ✅ Answer:
🔹 All sides of a rhombus are equal
🔹 Diagonals divide it into congruent triangles
🔹 Hence, diagonals bisect at right angles

🔒 ❓ Q28. Explain why an isosceles trapezium is not necessarily a parallelogram.
📌 ✅ Answer:
🔹 Only one pair of sides is parallel
🔹 A parallelogram needs two pairs of parallel sides
🔹 Hence, it fails the condition

🔒 ❓ Q29. Show logically that the sum of angles of an irregular quadrilateral is also 360°.
📌 ✅ Answer:
🔹 Draw a diagonal inside the quadrilateral
🔹 Two triangles are formed
🔹 Each has sum 180°, total 360°

🔒 ❓ Q30. Explain why a kite cannot have diagonals that bisect each other unless it is a rhombus.
📌 ✅ Answer:
🔹 In a kite, only one diagonal bisects the other
🔹 Mutual bisection requires equal opposite sides
🔹 This happens only in a rhombus

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