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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 opposite sides are parallel and equal in length. In parallelogram ABCDABCD, side AB∥CDAB \parallel CD and BC∥DABC \parallel DA. Also, AB=CDAB = CD and BC=DABC = DA.

A parallelogram showing opposite sides AB and CD are parallel and equal.
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The opposite angles of a parallelogram are equal. For example, ∠A=∠C\angle A = \angle C and ∠B=∠D\angle B = \angle D.

Parallelogram with opposite angles marked equal.
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Any two adjacent angles in a parallelogram are supplementary, meaning their sum is 180∘180^\circ. So, ∠A+∠B=180∘\angle A + \angle B = 180^\circ, ∠B+∠C=180∘\angle B + \angle C = 180^\circ, etc.

Adjacent angles in a parallelogram adding to 180 degrees.
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The diagonals of a parallelogram bisect each other. This means the point where the diagonals intersect is the midpoint of both diagonals.

Diagonals of a parallelogram bisecting each other at point O.

📐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.

Problem 3:

In the parallelogram ABCDABCD, the diagonals ACAC and BDBD intersect at point OO. If OA=5OA = 5 cm and OB=6OB = 6 cm, find the lengths of OCOC and ODOD.

Parallelogram ABCD with diagonals intersecting at O, showing lengths 5 and 6.

Solution:

  1. We know that the diagonals of a parallelogram bisect each other.
  2. Therefore, OA=OCOA = OC and OB=ODOB = OD.
  3. Given OA=5OA = 5 cm, then OC=5OC = 5 cm.
  4. Given OB=6OB = 6 cm, then OD=6OD = 6 cm.

Explanation:

Since diagonals bisect each other, the intersection point OO is the midpoint. Thus, each diagonal is split into two equal segments.

Problem 4:

In parallelogram HELPHELP, the lengths are given as OE=4OE = 4 units and HLHL is 55 more than PEPE. If OO is the point where diagonals intersect, find OHOH.

Parallelogram HELP with diagonals intersecting at O, with OE = 4.

Solution:

  1. In parallelogram HELPHELP, diagonals PEPE and HLHL bisect each other at OO.
  2. PE=2×OE=2×4=8PE = 2 \times OE = 2 \times 4 = 8 units.
  3. Given HL=PE+5HL = PE + 5, so HL=8+5=13HL = 8 + 5 = 13 units.
  4. Since OO is the midpoint of HLHL, OH=12×HL=132=6.5OH = \frac{1}{2} \times HL = \frac{13}{2} = 6.5 units.

Explanation:

We use the property that diagonals bisect each other to find the full length of one diagonal, then use the given relationship to find the other, and finally halve it to find the segment length.