Practice Questions on Properties of Parallelograms (original) (raw)

Last Updated : 10 Feb, 2026

A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. This simple definition leads to many interesting properties and formulas essential for solving geometric problems.

Solved Questions

**Example 1: In a parallelogram ABCD, if AB = 9 cm, BC = 7 cm, and the height corresponding to base AB is 5 cm, find the area of the parallelogram.

**Solution:

Area = Base × Height

= 9 cm × 5 cm
= 45 cm2

Area of Parallelogram is 45 cm2.

**Example 2: In parallelogram PQRS, if ∠P = 70°, find the measure of ∠Q.

**Solution:

Since consecutive angles in a parallelogram are supplementary,

∠P + ∠Q = 180o
= 70 + ∠Q = 180o

∠Q = 110o

**Example 3: In the parallelogram ABCD, determine the length of AC if the diagonals BD and AC connect at point O and AO = 5 cm.

**Solution:

A parallelogram's diagonals intersect one another, Thus

AC = 2 × AO

= 2 × 5 cm
= 10 cm

**Example 4: Determine the perimeter of the parallelogram WXYZ if WX = 12 cm and XY = 9 cm.

**Solution:

Perimeter of parallelogram = 2 × (base + side)

= 2 × (12 cm + 9 cm)
= 2 × 21 cm
= 42 cm

42 cm is the perimeter of parallelogram.

**Example 5: In a rectangle, one of the special types of parallelograms, if the length is 15 cm and the width is 10 cm, find the length of the diagonal.

**Solution:

Using the Pythagorean theorem:

**Diagonal = √{(length) 2 +( width) 2 }

= √ 152 + 102
= √ 225 + 100 = √325
= 5√13 cm

**Example 6: Determine the length of the other diagonal of a rhombus, a unique kind of parallelogram, where each side is 13 cm and one diagonal is 10 cm.

**Solution:

Let d1 and d2 be the diagonals

Given that a rhombus's diagonals bisect each other at right angles,
d12 + d22 = 4 × side2

⇒ 102 + d22 = 4 × 132
⇒ 100 + d22 = 676
⇒ d22 = 576
⇒ d2 = 24 cm

**Example 7: In parallelogram ABCD, if AB = 10 cm, AD = 6 cm, and the angle between them is 60°, find the area of the parallelogram

**Solution:

Area = base × height

⇒ Area = AB × AD × Sin60o
= 10 × 6 × √3/2
= 30√3 cm2

**Example 8: In parallelogram PQRS, if PQ = 5x - 7, QR = 2x + 3, and PQ = QR, find the value of x.

**Solution:

5x − 7 = 2x + 3

⇒ 5x - 2x = 3 + 7
⇒ 3x = 10
⇒ x = 10/3

**Example 9: In parallelogram ABCD, if angle A = 3x + 10° and angle C = 2x + 30°, find the value of x.

**Solution:

Since opposite angles in a parallelogram are equal,

3x + 10 = 2x + 10
⇒ 3x - 2x = 30 - 10
⇒ x = 20

**Example 10: In parallelogram ABCD, if the diagonals intersect at right angles and the lengths of the diagonals are 12 cm and 16 cm, find the area of the parallelogram.

**Solution:

Area of a parallelogram can also be calculated using the lengths of the diagonals if they intersect at right angles:
Area = 1/2 × d1 × d2

= 1/2 × 12 × 16
= 96 cm2

Properties of Parallelograms Practice Worksheet

Worksheet on properties of parallelogram is added on form of image added below:

**Question 1. Determine the area of the parallelogram ABCD if AB = 10 cm and the height corresponding to AB is 7 cm.

**Question 2. If ∠P = 120° in the parallelogram PQRS, determine the measure of ∠Q.

**Question 3. Determine the parallelogram’s perimeter in WXYZ if WX = 15 cm and XY = 10 cm.

**Question 4. Determine the length of AC in the parallelogram ABCD if the diagonals AC and BD cross at point O and AO = 6 cm.

**Question 5. Determine the length of the diagonal of a rectangle, a particular kind of parallelogram, if its length is 20 cm and its width is 15 cm.

**Question 6. If one diagonal in a rhombus, a different kind of parallelogram, is 12 cm long and each side is 10 cm, calculate the other diagonal length.

**Question 7. If PQ = 4x − 5, QR = 3x + 2, and PQ = QR in the parallelogram PQRS, then determine the value of x.

**Question 8. The value of x in the parallelogram ABCD is given by angles
A = 4x + 15° and C = 2x + 25°.

**Question 9. Calculate the area of the parallelogram WXYZ if the diagonals connect at a straight angle and have lengths of 10 and 14 cm.

**Question 10. Determine the area of the parallelogram ABCD if the angles between AB and AD are 45° and AB and AD are 8 and 6 cm, respectively.

**Answer Key: