Practice Questions on Properties of Parallelograms

Properties of Parallelograms: 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.
These quadrilaterals stand out from other forms due to their special qualities and traits. This geometric figure has successive angles that are supplementary and opposite angles that are equal, adding up to 180^o. The sum of all interior angles of a parallelogram is 360^o. A parallelogram's diagonals split the form in half, creating two congruent triangles.

Important Properties of Parallelogram

Opposite Sides are Equal: In a parallelogram, opposite sides are of equal length. AB = CD and AD = BC

Opposite Angles are Equal: Opposite angles in a parallelogram are equal. ∠A = ∠C and ∠B = ∠D

Consecutive Angles are Supplementary: The sum of the angles on the same side is 180 degrees. ∠A + ∠B = 180^oand ∠C + ∠D = 180^o

Diagonals Bisect Each Other: The diagonals of a parallelogram bisect each other. AO = OC and BO = OD

where O is the point of intersection of the diagonals.

Area of a Parallelogram(Area Computation): The area can be calculated using the base and height. Area = Base × Height

Parallelogram Law: The sum of the squares of the lengths of all sides equals the sum of the squares of the lengths of the diagonals. 2(AB² + AD²) = AC² + BD²

Perimeter of a Parallelogram: The total of all the sides makes up the perimeter of a parallelogram. D = 2 × (Base + Side)

Diagonals of a Parallelogram: The diagonals bisect each other, creating two congruent triangles.

Angles in a Parallelogram: Opposite angles are equal, and consecutive angles are supplementary.

Properties of Special Parallelograms

• Rectangle: The diagonals are equal and all angles are 90 degrees.
• Rhombus: The diagonals cut each other at right angles, and all of the sides are equal.
• Square: diagonals are equal and bisect each other at right angles; all sides are equal; all angles are 90 degrees.

Examples on Properties of parallelograms

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 = 180^o
= 70 + ∠Q = 180^o

∠Q = 110^o

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 × Sin60^o
= 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:

Answer Key:

• Ans 1. 70 cm²
• Ans 2. 60^o
• Ans 3. 50 cm
• Ans 4. 12 cm
• Ans 5. 25 cm
• Ans 6. 16 cm
• Ans 7. 7
• Ans 8. 20
• Ans 9. 70 cm²
• Ans 10. 24 cm²

Read More:

• Properties of Parallelograms
• Parallelogram Formulas
• Area of Parallelogram