Parallelogram: Opposite Sides Equal (Motivate) MCQs Quiz | Class 9
This Class IX Mathematics (Code 041) quiz focuses on Unit IV: Geometry. The topic covers the motivation that opposite sides of a parallelogram are equal and its converse. Test your understanding of these fundamental properties, review your score, and download the solution PDF for offline study.
Overview of Parallelogram Properties
In Class IX Geometry (Unit IV), one of the fundamental shapes studied is the parallelogram. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. However, there are specific properties related to the lengths of these sides that are crucial for solving geometric proofs and problems.
Key Theorem: Opposite Sides are Equal
The motivating theorem states: In a parallelogram, opposite sides are equal.
If ABCD is a parallelogram, then:
- Length of AB = Length of CD
- Length of AD = Length of BC
This property can be proved using triangle congruence (usually SAS or ASA criteria) by drawing a diagonal.
The Converse Theorem
The converse is also true and is a standard method to prove that a quadrilateral is a parallelogram. It states: If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
If in a quadrilateral ABCD, AB = CD and AD = BC, then ABCD is a parallelogram.
Important Formulas and Relations
| Concept | Formula / Relation |
|---|---|
| Opposite Sides | Side 1 = Opposite Side 1 Side 2 = Opposite Side 2 |
| Perimeter | 2 x (Sum of adjacent sides) |
| Opposite Angles | Angle A = Angle C Angle B = Angle D |
Quick Revision Points
- A square, rectangle, and rhombus are all specific types of parallelograms; therefore, their opposite sides are always equal.
- Knowing the perimeter and one side allows you to find the adjacent side using the formula: P = 2(a + b).
- If you are given algebraic expressions for opposite sides, equate them to solve for the variable.
Extra Practice Questions
- If the perimeter of a parallelogram is 40 cm and one side is 12 cm, find the length of the adjacent side.
- In parallelogram PQRS, if PQ = 3x – 5 and RS = x + 7, find the value of x.
- Prove that a quadrilateral is a parallelogram if its opposite sides are equal (Converse theorem proof outline).
- If the ratio of two adjacent sides is 1:2 and the perimeter is 30 cm, find the lengths of all sides.
- Can a trapezium have opposite sides equal? (Discuss isosceles trapezium versus parallelogram properties).
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