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Understanding Quadrilaterals - Properties of a Parallelogram

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A parallelogram is a quadrilateral where both pairs of opposite sides are parallel. Visually, it looks like a tilted rectangle where the opposite sides never meet even if extended indefinitely.

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Property of Sides: In a parallelogram, the opposite sides are of equal length. For a parallelogram ABCDABCD, this means AB=CDAB = CD and BC=DABC = DA. You can visualize this as two pairs of identical parallel sticks forming the boundary.

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Property of Angles: The opposite angles of a parallelogram are equal. For example, if you look at the diagonal corners, ∠A=∠C\angle A = \angle C and ∠B=∠D\angle B = \angle D.

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Adjacent Angles: Any two consecutive angles (angles sharing a common side) are supplementary, meaning their sum is 180∘180^\circ. Visually, if one corner is sharp (acute), the neighboring corner will be blunt (obtuse) to compensate, such that ∠A+∠B=180∘\angle A + \angle B = 180^\circ.

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Diagonals Property: The diagonals of a parallelogram bisect each other. If you draw two lines connecting opposite vertices, they will cross at a point that is the exact midpoint for both lines. Note that while they bisect each other, the diagonals themselves are not necessarily equal in length unless the shape is a rectangle.

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Angle Sum Property: Like every quadrilateral, the sum of all four interior angles in a parallelogram is always 360∘360^\circ. This can be visualized by splitting the parallelogram into two triangles using a diagonal, where each triangle contributes 180∘180^\circ.

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Area and Base-Height Relationship: The area is determined by the base and the perpendicular height. The height is the shortest vertical distance between the base and the opposite parallel side, forming a 90∘90^\circ angle with the base.

📐Formulae

Perimeter of a parallelogram = 2×(a+b)2 \times (a + b), where aa and bb are the lengths of adjacent sides.

Area of a parallelogram = Base×Height=b×hBase \times Height = b \times h

Sum of adjacent angles: ∠A+∠B=180∘\angle A + \angle B = 180^\circ

Opposite sides: Side1=SideoppositeSide_{1} = Side_{opposite}

Opposite angles: ∠A=∠C\angle A = \angle C and ∠B=∠D\angle B = \angle D

💡Examples

Problem 1:

In a parallelogram PQRSPQRS, the measure of ∠P=70∘\angle P = 70^\circ. Find the measures of the remaining three angles ∠Q\angle Q, ∠R\angle R, and ∠S\angle S.

Solution:

Step 1: Use the property that opposite angles are equal. Therefore, ∠R=∠P=70∘\angle R = \angle P = 70^\circ. Step 2: Use the property that adjacent angles are supplementary. So, ∠P+∠Q=180∘\angle P + \angle Q = 180^\circ. Step 3: Substitute the value of ∠P\angle P: 70∘+∠Q=180∘  ⟹  ∠Q=180∘−70∘=110∘70^\circ + \angle Q = 180^\circ \implies \angle Q = 180^\circ - 70^\circ = 110^\circ. Step 4: Use the opposite angle property again for ∠S\angle S. ∠S=∠Q=110∘\angle S = \angle Q = 110^\circ.

Explanation:

We applied two fundamental properties: opposite angles are equal and adjacent angles sum to 180∘180^\circ to find all unknown interior angles.

Problem 2:

The perimeter of a parallelogram is 3030 cm. If the longer side measures 99 cm, find the length of the shorter side.

Solution:

Step 1: Let the longer side be a=9a = 9 cm and the shorter side be bb. Step 2: Use the perimeter formula: Perimeter=2(a+b)Perimeter = 2(a + b). Step 3: Substitute the known values: 30=2(9+b)30 = 2(9 + b). Step 4: Divide by 22: 15=9+b15 = 9 + b. Step 5: Solve for bb: b=15−9=6b = 15 - 9 = 6 cm.

Explanation:

Since opposite sides of a parallelogram are equal, the perimeter is simply twice the sum of two adjacent sides. We used the algebraic equation P=2(a+b)P = 2(a+b) to solve for the missing side.