Properties of Parallelograms Practice Worksheet - GeeksforGeeks - Free Printable

Here is the complete solution to all 10 questions from the worksheet “Properties of Parallelograms”, with clear explanations for each.

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Q1. Determine the area of the parallelogram ABCD if AB = 10 cm and the height corresponding to AB is 7 cm.

Solution:
Area of a parallelogram = base × height
Base = AB = 10 cm
Height = 7 cm

→ Area = 10 × 7 = 70 cm²

✔ Answer: 70 cm²

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Q2. If ∠P = 120° in the parallelogram PQRS, determine the measure of ∠Q.

Solution:
In a parallelogram, consecutive angles are supplementary (add up to 180°).
So, ∠P + ∠Q = 180°
Given ∠P = 120° → ∠Q = 180° – 120° = 60°

✔ Answer: 60°

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Q3. Determine the parallelogram's perimeter in WXYZ if WX = 15 cm and XY = 10 cm.

Solution:
In a parallelogram, opposite sides are equal.
So, WX = YZ = 15 cm, and XY = WZ = 10 cm.
Perimeter = sum of all sides = 2×(WX + XY) = 2×(15 + 10) = 2×25 = 50 cm

✔ Answer: 50 cm

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Q4. Determine the length of AC in the parallelogram ABCD if the diagonals AC and BD cross at point O and AO = 6 cm.

Solution:
In a parallelogram, diagonals bisect each other.
So, AO = OC → AC = AO + OC = 6 + 6 = 12 cm

✔ Answer: 12 cm

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Q5. 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.

Solution:
A rectangle’s diagonal can be found using the Pythagorean Theorem:
Diagonal = √(length² + width²) = √(20² + 15²) = √(400 + 225) = √625 = 25 cm

✔ Answer: 25 cm

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Q6. 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.

Solution:
In a rhombus, diagonals bisect each other at right angles and also bisect the vertex angles.
Let diagonals be d₁ = 12 cm, d₂ = ?
Each half-diagonal forms a right triangle with the side of the rhombus.

Half of d₁ = 6 cm
Let half of d₂ = x
Then by Pythagoras:
x² + 6² = 10² → x² + 36 = 100 → x² = 64 → x = 8
So full diagonal d₂ = 2×8 = 16 cm

✔ Answer: 16 cm

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Q7. If PQ = 4x - 5, QR = 3x + 2, and PQ = QR in the parallelogram PQRS, then determine the value of x.

Solution:
In a parallelogram, opposite sides are equal, but here it says PQ = QR, which implies adjacent sides are equal → this makes it a rhombus (a special parallelogram).

Set PQ = QR:
4x – 5 = 3x + 2
Subtract 3x from both sides: x – 5 = 2
Add 5: x = 7

✔ Answer: x = 7

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Q8. The value of x in the parallelogram ABCD is given by angles A = 4x + 15° and C = 2x + 25°.

Solution:
In a parallelogram, opposite angles are equal.
So, ∠A = ∠C
→ 4x + 15 = 2x + 25
Subtract 2x: 2x + 15 = 25
Subtract 15: 2x = 10 → x = 5

✔ Answer: x = 5

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Q9. Calculate the area of the parallelogram WXYZ if the diagonals connect at a straight angle and have lengths of 10 and 14 cm.

Solution:
If diagonals intersect at a straight angle (i.e., 90°), then the parallelogram is a rhombus or kite, but since it’s specified as a parallelogram, it must be a rhombus (diagonals perpendicular ⇒ rhombus).

Area of a rhombus when diagonals are perpendicular:
Area = (d₁ × d₂) / 2 = (10 × 14) / 2 = 140 / 2 = 70 cm²

✔ Answer: 70 cm²

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Q10. 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.

Solution:
The area of a parallelogram can also be calculated using two adjacent sides and the included angle:
Area = AB × AD × sin(θ)
Where θ is the angle between them.

AB = 8 cm, AD = 6 cm, θ = 45°
sin(45°) = √2 / 2 ≈ 0.7071

Area = 8 × 6 × sin(45°) = 48 × (√2 / 2) = 24√2 cm²
You may leave it exact or approximate:

Exact: 24√2 cm²
Approximate: 24 × 1.4142 ≈ 33.94 cm²

✔ Answer: 24√2 cm² (or approx. 33.94 cm²)

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## ✔ Final Answers Summary:

1. 70 cm²
2. 60°
3. 50 cm
4. 12 cm
5. 25 cm
6. 16 cm
7. x = 7
8. x = 5
9. 70 cm²
10. 24√2 cm²

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