Parallelogram Test (Motivate) MCQs Quiz | Class 9
Class IX Mathematics (Code 041) Unit IV: Geometry. This quiz tests your understanding of the sufficient condition for a quadrilateral to be a parallelogram: when one pair of opposite sides is both parallel and equal. Answer the questions below and download the PDF solution.
Understanding the Parallelogram Test
In Class IX Geometry, specifically within the study of Quadrilaterals, a key theorem states: A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel. This is a powerful tool because it allows us to prove a shape is a parallelogram using only one pair of sides, rather than checking both pairs.
Key concepts covered in this quiz:
- Sufficiency: Knowing that AB = CD and AB || CD is enough to prove ABCD is a parallelogram.
- Proof Logic: This property is typically proved using triangle congruence (SAS rule) by drawing a diagonal.
- Implications: Once this condition is met, the other properties of a parallelogram (opposite angles equal, diagonals bisecting each other) automatically apply.
| Condition Known | Is it a Parallelogram? |
|---|---|
| One pair parallel only | No (Could be Trapezium) |
| One pair equal only | No (Could be Isosceles Trapezium) |
| One pair parallel AND equal | Yes |
| Both pairs parallel | Yes (Definition) |
Quick Revision Points
- If a quadrilateral has one pair of opposite sides equal and parallel, the other pair is automatically equal and parallel.
- This test minimizes the steps required in geometric proofs.
- The diagonal divides such a quadrilateral into two congruent triangles.
Extra Practice Questions
- Draw a quadrilateral ABCD where AB = 6cm and CD = 6cm. Adjust the angles until AB is parallel to CD. Measure AD and BC. Are they equal?
- Prove that the line segments joining the midpoints of the opposite sides of a quadrilateral bisect each other.
- If ABCD is a parallelogram and P and Q are midpoints of opposite sides AB and CD, prove that APCQ is a parallelogram using the “one pair opposite sides parallel and equal” test.
- Can a trapezium ever satisfy the condition of one pair of opposite sides being both parallel and equal? (Why/Why not?)
- In triangle ABC, D and E are midpoints of AB and AC. Extend DE to F such that DE = EF. Join CF. Prove BDFC is a parallelogram.
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