Quadrilateral | Maths Grade 9

Quadrilateral - Sub Topics

Understanding quadrilaterals and their classifications is essential for anyone working in fields that involve reasoning and mathematical analysis. From the design of buildings to the development of computer graphics, the significance of quadrilaterals extends across various disciplines. In this chapter, we will go through the parallelogram, square, rectangle, rhombus, trapezium and kite.

Solved Questions on Quadrilateral

Quadrilaterals

A quadrilateral is a polygon with four straight sides and four vertices or corners. The term "quadrilateral" is derived from Latin where “quadri” means “four” and “latus” means “side”. In simple terms, it is a four-sided figure enclosed by four straight segments or edges, forming four angles.

There are different types of quadrilaterals. Each quadrilateral has its own distinctive characteristics.

Parallelogram

Definition: A parallelogram is a quadrilateral whose opposite sides are both parallel.
Properties:
i. Opposite sides are parallel and equal in length.
ii. Opposite angles are equal.
iii. Diagonals bisect each other.
iv. Sum of adjacent angles is 180°.

Rectangle

Definition: A rectangle is a parallelogram whose opposite sides are equal and all angles are right angles.
Properties:
i. Opposite sides are parallel and equal in length.
ii. All angles are right angles.
iii. Diagonals are equal and bisect each other.
iv. Sum of adjacent angles is 180°.

Rhombus

Definition: A rhombus is a parallelogram with adjacent sides equal.
Properties:
i. All sides are equal.
ii. Opposite sides are parallel.
iii. Opposite angles are equal.
iv. Diagonals bisect each other and intersect at right angles (Diagonals are not equal and perpendicular to each other).
v. Sum of adjacent angles is 180°.

Square

Definition: A square is a special parallelogram with all sides equal and all angles at 90°.
Properties:
i. All sides are equal.
ii. Opposite sides are parallel.
iii. Each angle is a right angle.
iv. Diagonals are equal and intersect at right angles (Diagonals are perpendicular to each other).
v. Sum of adjacent angles is 180°.

Kite

Definition: A kite is a quadrilateral whose two pairs of adjacent sides are equal.
Properties:
i. Adjacent sides are equal in length.
ii. The sides opposite each other (opposite sides) are not equal in length.
iii. One pair of opposite angles is equal.
iv. The diagonals of a kite meet each other at right angles.
v. Diagonals bisect the angles of a kite.

Trapezium

Definition: A trapezium is a quadrilateral with only one pair of opposite sides parallel.
Properties:
i. A trapezium has a pair of parallel sides known as the bases. The other two sides are not parallel and are called legs and they are typically of different lengths.
iii. The angles at the bases (parallel sides) are supplementary (180°).
iv. The perpendicular distance between the bases is known as the height or altitude.

Isosceles Trapezium

Definition: An isosceles trapezium is a special trapezium whose one pair of opposite sides is parallel and the non-parallel sides (legs) are equal.
Properties:
i. The angles at the bottom (base angles) of this shape are the same.
ii. The diagonals are of equal length.

Understanding these shapes and their properties is a crucial part of geometry and various ways to make geometry simpler and more practical.

Summary & AI Mentor

Worksheet on Quadrilateral

Solved Questions on Quadrilateral

1. Which of the following shows the correct values of unknown angles?

a) x = 70°; y = 120°; z = 75°
b) x = 75°; y = 120°; z = 70°
c) x = 85°; y = 120°; z = 70°
d) x = 70°; y = 120°; z = 85°

Answer: d) x = 70°; y = 120°; z = 85°

Explanation: In trapezium ABCF, AB // CF

x + 110° = 180° [Co-interior angle, AB // CF]

∴ x = 70°

y + 60° = 180° [Co-interior angle, AB // CF]

∴ y = 120°

Since, ∠EFC = x = 70° [Alternate angle, CB || EF]

Sum of interior angles of a quadrilateral DEFC = 360°

⇒ z + 130° + ∠EFC + 75° = 360°
⇒ z + 130° + 70° + 75° = 360°
⇒ z + 275° = 360°
⇒ z = 360° − 275°
∴ z = 85°

Hence, x = 70°, y = 120° and z = 85°.

2. In the adjoining figure, ABCD is a rhombus and CDEF is a square. If ∠ABC = 62°, then what is the size of ∠AEF?

a) 56°
b) 66°
c) 76°
d) 86°

Answer: c) 76°

Explanation: If ABCD is a rhombus, then AB = BC = CD = DA.
If CDEF is a square, then CD = DE = EF = FC.
Hence, AB = BC = CD = DA = DE= EF = FC.

∠ABC = 62° [ Given]

∠ADC = 62° [Opposite angles in a rhombus are equal.]

∠EDC = 90° [Each angle of a square is right angle.]

So, ∠EDA = ∠EDC + ∠ADC = 90° + 62° = 152°

In △AED,

∠DEA = ∠DAE [Angles opposite to equal sides are equal, AD = DE]

Sum of interior angles of a triangle = 180°
⇒ ∠EDA + ∠DEA + ∠DAE = 180°
⇒ 152° + ∠DEA + ∠DEA = 180°
⇒ 2∠DEA = 180° − 152°
⇒ 2∠DEA = 28°
∴ ∠DEA = 14°

∠DEF = 90° [Each angle of a square is right angle.]
⇒ ∠AEF + ∠DEA = 90°
⇒ ∠AEF + 14° = 90°
⇒ ∠AEF = 90° − 14°
∴ ∠AEF = 76°

3. If one angle of a rhombus is 60° and the length of a side is 6 cm, what is the length of the diagonal AC?

a) 3√2 cm
b) 3√3 cm
c) 6√2 cm
d) 6√3 cm

Answer: d) 6√3 cm

Explanation: ABCD is a rhombus.

AB = BC = CD = DA = 6 cm [Side of a rhombus.]
∠ A = ∠C = 60° [Opposite angles are equal.]
∠ B = ∠D = 180° − 60° = 120° [Sum of adjacent angles is 180°.]

In △BAD,
∠ A = 60° [Given]
∠ABD = 120°/2 = 60° [BD bisect ∠B.]
∠ADB = 120°/2 = 60° [BD bisect ∠D.]

So, △BAD is an equilateral triangle.

Then, AB = BD = DA = 6 cm

As we know the diagonals of a rhombus bisect each other at right angles.
AO = OC,
BO = OD = 6/2 = 3 cm
∠AOB = 90°

In ∠AOB,
OA² = AB² − OB² [By Pythagoras theorem]
⇒ OA² = 6² − 3²
⇒ OA² = 36 − 9
⇒ OA² = 27
⇒ OA = √27
∴ OA = 3√3 cm

AC = 2 × OA = 2 × 3√3 cm = 6√3 cm

∴ The length of the diagonal AC is 6√3 cm.

4. Which of the following quadrilaterals is obtained by joining the mid-points of an isosceles trapezium?

a) Rectangle
b) Rhombus
c) Square
d) Kite

Answer: b) Rhombus

Explanation: ABCD is an isosceles trapezium in which AB || CD and AD = BC. AC and BD are the diagonals of an isosceles trapezium ABCD.

P, Q, R and S are the midpoints of the sides AB, BC, CD and DA.
PQ, QR, RS and SP are joined to form the quadrilateral PQRS.

In an isosceles trapezium ABCD, the length of diagonals is equal.
AC = BD

Using the mid-point theorem,
In △ABC,
If P and Q are the midpoints of AB and BC, then PQ || AC and PQ = ½ AC......(I)
In △ADC,
If S and R are midpoints of CD and AD, then SR || AC and SR = ½ AC....(II)

From (I) and (II), PQ || SR and PQ = SR …………..(a)

In △ABD,
If P and S are the midpoints of AB and DA, then PS || BD and PS = ½ BD......(III)
In △BCD,
If Q and R are the midpoints of BC and CD, then QR || BD and QR = ½ BD....(IV)

From (III) and (IV), PS || QR and PS = QR …………..(b)

Thus, PQRS is a parallelogram.

AD = BC
AS = ½ AD = ½ BC = BQ

In △APS and △BPQ,
AP = BP [P is the midpoint.]
∠A = ∠B [Angles opposite to equal sides are equal, AD = BC]
AS = BQ [Found above.]
△APS ≅ △BPQ [By SAS axiom of congruency.]
PS = PQ [By C.P.C.T] …………..(c)

From (a), (b) and (c), the adjacent sides of a parallelogram ABCD are equal.

PQ = SR = PS = QR

No angles of an isosceles trapezium ABCD is 90°. So, PQRS has no right angle.

∴ Hence, all sides of parallelogram PQRS are equal. Therefore, PQRS is a rhombus.

5. What is the difference between ∠DIE and ∠TEO if ∠ITE and ∠IDE of a kite TIDE are 124° and 58°, respectively?

a) 13°
b) 23°
c) 33°
d) 63°

Answer: c) 33°

Explanation: The diagonals of a kite TIDE meet each other at right angles and bisect the angles.

∠IOT = ∠EOT= ∠DOE = ∠DOI = 90°

∠ITE = 124°
∠ITO = ∠ETO = ∠ITE/2 = 124°/2 = 62° [Diagonal TD bisects the angles ∠ITE.]

Similarly, ∠IDE = 58°
∠ODI = ∠ODE = ∠IDE/2 = 58°/2 = 29° [Diagonal TD bisects the angles ∠IDE.]

In △IOD,
Sum of interior angles of a triangle = 180°
⇒ ∠DIO + ∠DOI + ∠ETO = 180°
⇒ ∠DIO + 90° + 29° = 180°
⇒ ∠DIO = 61°
∴ ∠DIE = ∠DIO = 61°

In △TOE,
Sum of interior angles of a triangle = 180°
⇒ ∠TEO + ∠TOE + ∠ODI = 180°
⇒ ∠TEO + 90° + 62° = 180°
∴ ∠TEO = 28°

Difference between ∠DIE and ∠TEO = 61° − 28° = 33°

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