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Quadrilaterals

NCERT Class 8 · Mathematics Based on NCERT Class 8 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter explores four-sided figures called quadrilaterals, focusing on their properties, types, and relationships. The big idea is that all quadrilaterals have a sum of interior angles equal to 360°, and special quadrilaterals like rectangles, squares, parallelograms, rhombuses, kites, and trapeziums have unique properties that can be deduced using congruence and parallel line theorems. A student must walk away understanding how to identify, define, and apply the properties of each quadrilateral type.
Key Concepts (13)
Quadrilateral
A quadrilateral is a four-sided polygon. The word is derived from Latin words 'quadri' meaning four and 'latus' meaning sides.
e.g. A window pane, a book cover, or a chessboard square are all examples of quadrilaterals.
Rectangle
A rectangle is a quadrilateral in which all angles are right angles (90°).
e.g. A classroom blackboard or a smartphone screen is a rectangle.
Square
A square is a quadrilateral in which all angles are 90° and all sides are of equal length.
e.g. A chessboard square or a floor tile is a square.
Parallelogram
A parallelogram is a quadrilateral in which opposite sides are parallel.
e.g. The shape of a window grill or a tilted table top is a parallelogram.
Rhombus
A rhombus is a quadrilateral in which all sides have the same length.
e.g. A diamond-shaped kite or a rhombus-shaped earring is a rhombus.
Kite
A kite is a quadrilateral that can be labelled ABCD such that AB = BC and CD = DA.
e.g. A traditional Indian kite (patang) is a kite shape.
Trapezium
A trapezium is a quadrilateral with at least one pair of parallel opposite sides.
e.g. The shape of a bucket or a table lamp stand is a trapezium.
Isosceles Trapezium
An isosceles trapezium is a trapezium in which the non-parallel sides have the same length.
e.g. The shape of a road sign or a bridge support is an isosceles trapezium.
Diagonals
Diagonals are line segments joining opposite vertices of a quadrilateral.
e.g. In a rectangle, the two diagonals are equal in length and bisect each other.
Bisect
Bisecting a quantity means dividing it into two equal parts.
e.g. The diagonals of a rectangle bisect each other, meaning they cut each other into two equal halves at their intersection point.
Congruence
Congruence means two geometric figures are identical in shape and size. Triangles can be proven congruent using conditions like SAS, AAS, ASA, or SSS.
e.g. In a rectangle, triangle ADC is congruent to triangle DAB by SAS condition.
Venn Diagram
A Venn diagram is a diagram using circles or ovals to represent sets of objects, showing relationships between them.
e.g. The set of squares is inside the set of rectangles, which is inside the set of parallelograms.
Conjecture
A conjecture is a statement about which we are highly confident but not yet sure if it always holds true, based on observation.
e.g. Observing that a few rectangles have diagonals that bisect each other leads to a conjecture, but it must be proved.
Formulas & Equations (2)
Sum of interior angles of a quadrilateral = 360°
Reads as: The sum of all interior angles of any quadrilateral is 360 degrees.
Variables: No variables; it is a constant property.
Use when: Use this to find a missing angle in any quadrilateral when three angles are known.
⚠ Students often forget that this applies to all quadrilaterals, not just rectangles or squares, and incorrectly use 180°.
In a triangle, sum of interior angles = 180°
Reads as: The sum of all interior angles of any triangle is 180 degrees.
Variables: No variables; it is a constant property.
Use when: Use this to find missing angles in triangles formed by drawing a diagonal in a quadrilateral.
⚠ Students sometimes apply 360° to triangles instead of 180°.
How It Works
Deducing that diagonals of a rectangle are equal
  1. Consider rectangle ABCD with diagonals AC and BD.
  2. In triangles ADC and DAB, note that AB = CD (opposite sides of rectangle), ∠BAD = ∠CDA = 90°, and AD is common.
  3. By SAS congruence condition, ∆ADC ≅ ∆DAB.
  4. Therefore, AC = BD (corresponding parts of congruent triangles).
💡 The SAS condition requires two sides and the included angle to be equal. Students often forget to check the included angle.
Deducing that diagonals of a rectangle bisect each other
  1. Consider rectangle ABCD with diagonals intersecting at O.
  2. Show that ∠1 = ∠2 using angle sum in triangles (∠1 = 90° – ∠3, ∠2 = 90° – ∠3).
  3. In triangles AOB and COD, ∠1 = ∠2, ∠AOB = ∠COD (vertically opposite), and AB = CD (opposite sides).
  4. By AAS condition, ∆AOB ≅ ∆COD.
  5. Therefore, OA = OC and OB = OD, so O is the midpoint of both diagonals.
💡 Students often confuse AAS with ASA. Ensure the order of angles and side is correct.
Deducing that a quadrilateral with all angles 90° is a rectangle
  1. Consider quadrilateral ABCD with all angles 90°.
  2. Join diagonal BD.
  3. In triangles BAD and DCB, ∠BAD = ∠DCB = 90°, BD is common, and ∠1 = ∠2 (proved using angle sum).
  4. By AAS condition, ∆BAD ≅ ∆DCB.
  5. Therefore, AD = CB and DC = BA, so opposite sides are equal, making it a rectangle.
💡 Students often think that having all angles 90° automatically means opposite sides are equal, but this must be proved.
Deducing that diagonals of a square bisect each other at right angles
  1. Consider square ABCD with diagonals intersecting at O.
  2. In triangles BOA and BOC, AB = BC (sides of square), BO is common, and AO = OC (diagonals bisect each other).
  3. By SSS condition, ∆BOA ≅ ∆BOC.
  4. Therefore, ∠BOA = ∠BOC (corresponding parts).
  5. Since ∠BOA + ∠BOC = 180° (straight line), each is 90°.
💡 Students often forget that the diagonals of a square are equal and bisect each other at 90°, but they are not necessarily equal in a rhombus.
Deducing that opposite angles of a parallelogram are equal
  1. Consider parallelogram ABCD with AB || CD and AD || BC.
  2. Since AB || CD, ∠A + ∠D = 180° (interior angles on same side of transversal AD).
  3. Since AD || BC, ∠A + ∠B = 180°.
  4. Therefore, ∠D = ∠B (both equal 180° – ∠A).
  5. Similarly, ∠A = ∠C.
💡 Students often forget that adjacent angles in a parallelogram are supplementary, not equal.
Deducing that opposite sides of a parallelogram are equal
  1. Consider parallelogram ABCD with diagonal BD.
  2. In triangles ABD and CDB, ∠ABD = ∠CDB (alternate angles, AB || CD), ∠ADB = ∠CBD (alternate angles, AD || BC), and BD is common.
  3. By ASA condition, ∆ABD ≅ ∆CDB.
  4. Therefore, AD = CB and AB = CD.
💡 Students often use the wrong congruence condition. Ensure the correct pair of angles and side is used.
Deducing that diagonals of a parallelogram bisect each other
  1. Consider parallelogram EASY with diagonals intersecting at O.
  2. In triangles AOE and YOS, AE = YS (opposite sides), ∠OAE = ∠OYS (alternate angles), and ∠OEA = ∠OSY (alternate angles).
  3. By ASA condition, ∆AOE ≅ ∆YOS.
  4. Therefore, OA = OY and OE = OS, so O is the midpoint of both diagonals.
💡 Students often forget that the diagonals of a parallelogram are not necessarily equal, only that they bisect each other.
Deducing that diagonals of a rhombus bisect each other at right angles
  1. Consider rhombus GAME with diagonals intersecting at O.
  2. In triangles GEO and MEO, GE = ME (sides of rhombus), GO = MO (diagonals bisect each other), and EO is common.
  3. By SSS condition, ∆GEO ≅ ∆MEO.
  4. Therefore, ∠GOE = ∠MOE (corresponding parts).
  5. Since ∠GOE + ∠MOE = 180°, each is 90°.
💡 Students often confuse this with the property of a square. In a rhombus, diagonals are not necessarily equal, only perpendicular.
Deducing that diagonals of a rhombus bisect its angles
  1. Consider rhombus GAME with diagonal AE.
  2. In triangles GAE and MAE, GA = MA (sides of rhombus), GE = ME (sides of rhombus), and AE is common.
  3. By SSS condition, ∆GAE ≅ ∆MAE.
  4. Therefore, ∠GAE = ∠MAE and ∠GEA = ∠MEA, so diagonal AE bisects the angles.
💡 Students often forget that this property applies to both diagonals of a rhombus.
Deducing that in an isosceles trapezium, base angles are equal
  1. Consider isosceles trapezium UVWX with UV || XW and UX = VW.
  2. Draw perpendiculars XY and WZ to UV, forming rectangle XWZY.
  3. In triangles UXY and VWZ, UX = VW (given), XY = WZ (opposite sides of rectangle), and ∠UXY = ∠VWZ = 90°.
  4. By RHS condition, ∆UXY ≅ ∆VWZ.
  5. Therefore, ∠U = ∠V.
💡 Students often forget that the perpendiculars create right angles, which are necessary for RHS congruence.
Exam Traps — Don't Make These Mistakes
⚠ Confusing rectangle and square definitions
✗ A rectangle has all sides equal.
→ A rectangle only requires opposite sides to be equal, not all four sides. That is the property of a square.
✓ A rectangle is a quadrilateral with all angles 90° and opposite sides equal.
⚠ Assuming all parallelograms are rectangles
✗ A parallelogram has all angles 90°.
→ A parallelogram only requires opposite sides to be parallel. Angles are not necessarily 90°; only in a rectangle are all angles 90°.
✓ A parallelogram is a quadrilateral with opposite sides parallel. Adjacent angles are supplementary, and opposite angles are equal.
⚠ Mixing up diagonal properties of rhombus and square
✗ The diagonals of a rhombus are equal.
→ In a rhombus, diagonals are perpendicular and bisect each other, but they are not necessarily equal. Only in a square are diagonals equal, perpendicular, and bisecting.
✓ The diagonals of a rhombus bisect each other at right angles, but they are not necessarily equal.
⚠ Forgetting the sum of angles in a quadrilateral
✗ The sum of angles in a quadrilateral is 180°.
→ The sum of angles in a triangle is 180°, but in a quadrilateral it is 360°.
✓ The sum of all interior angles in any quadrilateral is 360°.
⚠ Incorrectly applying congruence conditions
✗ ∆ABD ≅ ∆CDB by SAS.
→ Students often claim SAS without verifying that the angle is included between the two sides. For example, in a parallelogram, the correct condition is often ASA or AAS.
✓ Specify the correct condition: 'By ASA condition, ∆ABD ≅ ∆CDB because ∠ABD = ∠CDB, BD is common, and ∠ADB = ∠CBD.'
⚠ Thinking a kite is always a rhombus
✗ A kite has all sides equal.
→ A kite only has two pairs of adjacent equal sides, not all four sides equal. A rhombus has all four sides equal.
✓ A kite is a quadrilateral with AB = BC and CD = DA, but not necessarily all sides equal.
Compare & Contrast
Sides
Rectangle
Square
A rectangle has opposite sides equal, while a square has all four sides equal.
Angles
Rhombus
Square
A rhombus does not necessarily have 90° angles, while a square always has all angles 90°.
Angles
Parallelogram
Rectangle
A parallelogram has opposite angles equal but not necessarily 90°, while a rectangle has all angles 90°.
Sides
Kite
Rhombus
A kite has two pairs of adjacent equal sides, while a rhombus has all four sides equal.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Short Answer (3 marks) on properties, 1 Long Answer (5 marks) on proof or construction
Most asked: Properties of rectangles, squares, parallelograms, and rhombuses · Sum of angles in a quadrilateral (360°) · Diagonal properties (equal, bisect, perpendicular) · Congruence proofs for diagonal properties · Venn diagram relationships between quadrilaterals
Don't skip: Prove that the diagonals of a rectangle are equal and bisect each other.; Prove that the diagonals of a rhombus bisect each other at right angles.; Find missing angles in a quadrilateral using the 360° property.; Construct a quadrilateral given diagonal lengths and angle between them.
One-Line Revision (read aloud before the exam)
  1. A quadrilateral is a four-sided polygon; the sum of its interior angles is always 360°.
  2. A rectangle has all angles 90° and opposite sides equal.
  3. A square has all angles 90° and all sides equal.
  4. A parallelogram has opposite sides parallel and equal, and opposite angles equal.
  5. A rhombus has all sides equal, opposite sides parallel, and diagonals bisect each other at 90°.
  6. A kite has two pairs of adjacent equal sides.
  7. A trapezium has at least one pair of parallel opposite sides.
  8. An isosceles trapezium has non-parallel sides equal and base angles equal.
  9. The diagonals of a rectangle are equal and bisect each other.
  10. The diagonals of a square are equal, bisect each other at 90°, and bisect the angles.
  11. The diagonals of a rhombus bisect each other at 90° and bisect the angles.
  12. Every square is a rectangle, but not every rectangle is a square.
  13. Every square is a rhombus, but not every rhombus is a square.
  14. Every rectangle is a parallelogram, but not every parallelogram is a rectangle.
  15. Adjacent angles in a parallelogram are supplementary (sum to 180°).
  16. Congruence conditions (SAS, ASA, AAS, SSS) are used to prove diagonal properties.

MCQs — 10 Questions with Answers

Q1. According to the definition given in the chapter, which of the following is NOT a property of a rectangle?

  • A. All angles are right angles (90°)
  • B. Opposite sides are equal in length
  • C. Diagonals are perpendicular to each other ✓
  • D. Opposite sides are parallel

Answer: C — The definition of a rectangle requires all angles to be 90° and opposite sides to be equal, but the chapter proves that diagonals of a rectangle bisect each other and are equal in length, not necessarily perpendicular.

Q2. A carpenter has one wooden strip that is 12 cm long. According to the carpenter's problem in the chapter, what should be the length of the second strip to form a rectangle when both strips represent the diagonals?

  • A. 6 cm
  • B. 12 cm ✓
  • C. 24 cm
  • D. 18 cm

Answer: B — The chapter proves through congruence that the diagonals of a rectangle are always equal in length, so if one diagonal is 12 cm, the other must also be 12 cm.

Q3. The word 'quadrilateral' is derived from Latin words. What does 'latus' refer to?

  • A. Four
  • B. Sides ✓
  • C. Angles
  • D. Shape

Answer: B — According to the chapter, 'quadrilateral' comes from 'quadri' meaning four and 'latus' referring to sides.

Q4. In rectangle ABCD with diagonals AC and BD intersecting at point O, which of the following statements is true?

  • A. O is located at one-third of the diagonal from A
  • B. O is the midpoint of both diagonals AC and BD ✓
  • C. O divides the diagonals in the ratio 2:1
  • D. O is closer to vertices A and B than to C and D

Answer: B — Through the congruence proof (Deduction 2), the chapter establishes that OA = OC and OB = OD, making O the midpoint of both diagonals.

Q5. A farmer in Mozambique wants to construct the rectangular base of a house using two wooden sticks as diagonals. The sticks are 10 metres long each and are placed so they bisect each other. What property must these sticks have to guarantee the base will be rectangular?

  • A. They must be perpendicular to each other
  • B. They must form a 45° angle with each other
  • C. They must be equal in length and bisect each other at any angle ✓
  • D. They must intersect at exactly one-quarter of their length

Answer: C — The chapter proves through Deduction 3 that regardless of the angle between the diagonals, if they are equal and bisect each other, all angles of the quadrilateral will be 90° and opposite sides will be equal, forming a rectangle.

Q6. In the proof using triangles AOB and COD (Deduction 2), which congruence condition is used to prove that ∆AOB ≅ ∆COD?

  • A. SSS (Side-Side-Side)
  • B. SAS (Side-Angle-Side)
  • C. AAS (Angle-Angle-Side) ✓
  • D. RHS (Right Angle-Hypotenuse-Side)

Answer: C — The chapter shows that ∠1 = ∠2, the vertically opposite angles are equal, and AD = BC, so the AAS condition is used to prove the congruence of triangles AOB and COD.

Q7. When two diagonals of a rectangle intersect at point O, and the angle between them is x, what is the measure of each base angle in the isosceles triangle AOB?

  • A. x
  • B. 90 + x/2
  • C. 90 - x/2 ✓
  • D. 180 - x

Answer: C — Through the calculation in Deduction 3, since the sum of angles in triangle AOB is 180° and the angle at O is x, the two equal base angles sum to (180 - x), making each base angle equal to 90 - x/2.

Q8. Which of the following is the correct definition of 'bisect' as used in the chapter?

  • A. To divide into three equal parts
  • B. To divide into two equal parts ✓
  • C. To cut at right angles
  • D. To intersect at an angle of 60°

Answer: B — The chapter explicitly defines bisecting as dividing a quantity into two equal parts, which is why the diagonals bisecting each other means they divide each other into equal halves.

Q9. In rectangle ABCD, if we know that AC = BD (both diagonals are equal) and O is their intersection point with AO = OC and BO = OD, which statement would NOT necessarily be true?

  • A. All interior angles of ABCD are 90°
  • B. Opposite sides AB and CD are equal
  • C. The diagonals are perpendicular to each other ✓
  • D. Opposite sides AD and BC are equal

Answer: C — The chapter's Deduction 3 proves that all angles are 90° and opposite sides are equal regardless of the angle between diagonals, but diagonals being perpendicular is NOT required for a rectangle.

Q10. In the carpenter's problem, if the two wooden strips representing diagonals are placed such that they bisect each other but are NOT equal in length, which of the following best describes the quadrilateral formed?

  • A. It will always be a rectangle
  • B. It will be a parallelogram but not necessarily a rectangle ✓
  • C. It cannot form a valid quadrilateral
  • D. It will be a rectangle only if the angle between them is 90°

Answer: B — The chapter proves that for a rectangle, the diagonals must be BOTH equal AND bisect each other; if only the bisecting property holds but lengths are unequal, the resulting quadrilateral is a parallelogram with no guarantee of right angles.

Flashcards

What is the definition of a quadrilateral?

A quadrilateral is a four-sided polygon with four vertices and four angles.

State the simplest definition of a rectangle.

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

What does 'bisect' mean in geometry?

To bisect means to divide a line segment or angle into two equal parts.

What is the key property of diagonals in a rectangle?

The diagonals of a rectangle are equal in length and bisect each other at their midpoint.

Are the diagonals of a rectangle perpendicular to each other?

No, the diagonals of a rectangle can meet at any angle and still form a rectangle.

Can a quadrilateral with all 90° angles have unequal opposite sides?

No, if all angles are 90°, the opposite sides must be equal (proven by congruence).

What is the difference between a conjecture and a proof?

A conjecture is a confident guess based on observation; a proof is a logical deduction that always holds true.

Which congruence condition is used to prove ∆AOB ≅ ∆COD in a rectangle?

The AAS (Angle-Angle-Side) congruence condition is used.

If diagonals of a quadrilateral are equal and bisect each other, what shape must it be?

It must be a rectangle, regardless of the angle between the diagonals.

Why are opposite sides of a rectangle parallel?

Because when a transversal crosses two lines and co-interior angles sum to 180°, the lines are parallel.

Important Board Questions

Define a rectangle using only the property of its angles. [1 mark]

A rectangle is a quadrilateral in which ___. Use only the angle property, proven by congruence.

Write down any two properties of the diagonals of a rectangle. How do these properties help a carpenter make a rectangular frame? [2 marks]

Property 1: equal length. Property 2: bisect each other. Carpenters use these to position wooden strips as diagonals.

In rectangle ABCD, the diagonals AC and BD intersect at O. Prove that ∆AOB ≅ ∆COD using the AAS congruence condition. (You may refer to the angles ∠1 and ∠2 shown in the figure.) [3 marks]

Show that ∠1 = ∠2 using the angle sum property and right angles. Then use one common side and AAS congruence to prove the triangles are equal.

Two wooden strips of length 8 cm each are placed as the diagonals of a quadrilateral such that they bisect each other at point O. The angle between the diagonals is x°. Prove that all angles of the quadrilateral formed are 90°, regardless of the value of x. (Assume the four angles at O are x°, x°, (180−x)°, and (180−x)°.) [5 marks]

In isosceles ∆AOB, use a = (180−x)/2 = 90 − x/2. In isosceles ∆AOD, use b = x/2. Add a + b to show each angle of the quadrilateral equals 90°. Use congruence to verify opposite sides are equal.

Assertion-Reason Questions for Quadrilaterals

Q1. Assertion (A):

The diagonals of a rectangle are always equal in length.

Reason (R):

When triangles ADC and DAB are congruent by SAS (AB = CD, ∠BAD = ∠CDA = 90°, AD is common), their corresponding sides AC and BD must be equal.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true from the chapter's Deduction 1. The reason correctly explains why the assertion is true using the congruence condition SAS and the property of corresponding parts of congruent triangles.

Q2. Assertion (A):

If a quadrilateral has diagonals that bisect each other, then the quadrilateral must be a rectangle.

Reason (R):

Bisecting diagonals mean each diagonal cuts the other into two equal parts at the point of intersection.

Answer: D — A is false but R is true — The assertion is false—diagonals bisecting each other is necessary but not sufficient for a rectangle. A parallelogram also has bisecting diagonals. The reason correctly defines bisection but doesn't explain why the assertion would be true. The chapter proves that equal AND bisecting diagonals (together) form a rectangle.

Q3. Assertion (A):

All angles of a quadrilateral formed by two equal diagonals that bisect each other are 90°.

Reason (R):

When diagonals bisect each other, the triangles formed are all congruent to each other.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true, as proven in Deduction 3 of the chapter. However, the reason is not a complete explanation. The triangles are not all congruent to each other; rather, opposite triangles are congruent (∆AOB ≅ ∆COD, ∆AOD ≅ ∆COB). The 90° angles result from specific angle calculations, not just from congruence.

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Case-Based Questions (CBQ) for Quadrilaterals

Case Study:

A construction company is building a rectangular swimming pool. The architect provides two plans: Plan A where the diagonals measure 15 m each and meet at their exact midpoints, and Plan B where both diagonals measure 15 m but meet at a point 6 m from one end and 9 m from the other end of each diagonal. The construction supervisor needs to verify which plan will actually produce a rectangular pool.

Q1. Which plan satisfies the condition that diagonals bisect each other?

  • A. Plan A only ✓
  • B. Plan B only
  • C. Both plans satisfy this condition
  • D. Neither plan satisfies this condition

Answer: A

Q2. Based on the chapter's deductions, which plan will definitely produce a rectangle?

  • A. Plan A, because diagonals are equal and bisect each other ✓
  • B. Plan B, because diagonals are equal
  • C. Both plans will produce a rectangle
  • D. Neither plan will produce a rectangle

Answer: A

Q3. If Plan A is used and the diagonals intersect at point O, what is the distance from a corner to O?

  • A. 30 m
  • B. 15 m
  • C. 7.5 m ✓
  • D. Cannot be determined

Answer: C

Q4. What additional measurement should the supervisor take to verify the pool is rectangular using Plan A?

  • A. Measure all four angles to confirm they are 90°
  • B. Just measure the diagonal lengths
  • C. Measure that the diagonals meet at their midpoints ✓
  • D. Check that opposite sides are equal

Answer: C

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