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Class 9 Maths Chapter 8 Quadrilaterals – Exercise 8.2 NCERT Solutions

Class 9 Maths Chapter 8 Quadrilaterals – Exercise 8.2 NCERT Solutions

Introduction

Exercise 8.2 explores the properties of parallelograms. You’ll learn how to prove that opposite sides and angles are equal, diagonals bisect each other, and apply these properties to solve problems.

Key Concepts

  • Parallelogram Properties:

    • Opposite sides are equal.

    • Opposite angles are equal.

    • Diagonals bisect each other.

    • Each diagonal divides the parallelogram into two congruent triangles.

Solved Questions (Step by Step)

Q1. In parallelogram ABCD, prove AB = CD and AD = BC.

Solution:

  • Opposite sides of a parallelogram are equal.

  • Hence, AB = CD and AD = BC.

Q2. In parallelogram PQRS, prove ∠P = ∠R and ∠Q = ∠S.

Solution:

  • Opposite angles of a parallelogram are equal.

  • Hence, ∠P = ∠R and ∠Q = ∠S.

Q3. In parallelogram LMNO, prove diagonals bisect each other.

Solution:

  • Diagonals intersect at point X.

  • LX = XO and MX = XN.

  • Hence, diagonals bisect each other.

Q4. In parallelogram ABCD, prove ΔABC ≅ ΔCDA.

Solution:

  • Diagonal AC divides parallelogram into two congruent triangles.

  • Hence, ΔABC ≅ ΔCDA.

Q5. In parallelogram PQRS, prove ΔPQS ≅ ΔQRS.

Solution:

  • Diagonal QS divides parallelogram into two congruent triangles.

  • Hence, ΔPQS ≅ ΔQRS.

Q6. In parallelogram LMNO, prove ΔLMN ≅ ΔONM.

Solution:

  • Diagonal MN divides parallelogram into two congruent triangles.

  • Hence, ΔLMN ≅ ΔONM.

Q7. In parallelogram ABCD, prove AB ∥ CD and AD ∥ BC.

Solution:

  • By definition, opposite sides of a parallelogram are parallel.

Q8. In parallelogram PQRS, prove ∠P + ∠Q = 180°.

Solution:

  • Adjacent angles in a parallelogram are supplementary.

  • Hence, ∠P + ∠Q = 180°.

Q9. In parallelogram LMNO, prove ∠L + ∠M = 180°.

Solution:

  • Adjacent angles are supplementary.

  • Hence, ∠L + ∠M = 180°.

Q10. In parallelogram ABCD, prove ∠A + ∠B = 180°.

Solution:

  • Adjacent angles are supplementary.

  • Hence, ∠A + ∠B = 180°.

Q11. In parallelogram PQRS, prove ∠Q + ∠R = 180°.

Solution:

  • Adjacent angles are supplementary.

Q12. In parallelogram LMNO, prove ∠M + ∠N = 180°.

Solution:

  • Adjacent angles are supplementary.

Q13. In parallelogram ABCD, prove ∠C + ∠D = 180°.

Solution:

  • Adjacent angles are supplementary.

Q14. In parallelogram PQRS, prove ∠S + ∠P = 180°.

Solution:

  • Adjacent angles are supplementary.

Q15. In parallelogram LMNO, prove ΔLMO ≅ ΔNMO.

Solution:

  • Diagonal MO divides parallelogram into two congruent triangles.

FAQs (10 for Exercise 8.2)

  1. Q: What is a parallelogram? A: A quadrilateral with both pairs of opposite sides parallel.

  2. Q: What are the properties of a parallelogram? A: Opposite sides equal, opposite angles equal, diagonals bisect each other.

  3. Q: Are adjacent angles in a parallelogram supplementary? A: Yes, their sum is 180°.

  4. Q: Do diagonals of a parallelogram bisect each other? A: Yes.

  5. Q: Can a rectangle be considered a parallelogram? A: Yes, it is a special type of parallelogram.

  6. Q: Can a rhombus be considered a parallelogram? A: Yes, it is a special type of parallelogram.

  7. Q: What is the difference between a parallelogram and a trapezium? A: Parallelogram has two pairs of parallel sides; trapezium has one pair.

  8. Q: What is the sum of angles in a parallelogram? A: 360°.

  9. Q: How many diagonals does a parallelogram have? A: Two, and they bisect each other.

  10. Q: Why are parallelogram properties important? A: They help in proving geometric theorems and solving problems.

Conclusion

Exercise 8.2 has 15 questions that strengthen your understanding of parallelogram properties. This builds the foundation for studying special quadrilaterals like rectangles, rhombuses, and squares.

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