Class 9 Maths Chapter 8 Quadrilaterals NCERT Solutions: All Exercises Step-by-Step + Extra Practice

March 16, 2026

Class 9 Maths Chapter 8 Quadrilaterals NCERT Solutions: All Exercises Step-by-Step + Extra Practice

Introduction

Chapter 8 takes you from triangles to quadrilaterals - shapes with four sides where all interior angles always add up to 360 degrees. You'll learn parallelogram properties and the mid-point theorem that says joining midpoints of any quadrilateral creates a parallelogram.

Two exercises in NCERT: basic angle problems (8.1) and theorems/proofs (8.2). I've added extra questions that students often struggle with on exams.

Key Properties and Theorems

Quadrilateral angle sum: 360°
Parallelogram: Opposite sides equal and parallel, opposite angles equal, consecutive angles add to 180°, diagonals bisect each other
Mid-Point Theorem: Line segment joining midpoints of two sides of triangle is parallel to third side and half as long
Varignon's Theorem: Midpoints of any quadrilateral form parallelogram

Exercise 8.1 Solved Questions

Question 1: Angles of quadrilateral in ratio 3:5:9:13. Find measures.

Solution: Let angles be 3x, 5x, 9x, 13x. Sum: 30x = 360°. x = 12°. Angles: 36°, 60°, 108°, 156°.

Question 2: Parallelogram ABCD, angle D = 75°. Find angles B, C.

Solution: Opposite angles equal: angle B = angle D = 75°. Consecutive supplementary: angle A = 180° - 75° = 105°. Angle C = 105°.

Question 3: Can quadrilateral have 3 obtuse angles?

Solution: No. Three obtuse (>90°) would sum >270°. Leaves <90° for fourth angle, impossible since all positive.

Question 4: Sides AB parallel CD, angle DAB = angle BCD = 110°. Prove ABCD parallelogram.

Solution: Alternate interior angles equal (AB||CD). So angle ABC = angle BCD = 110°. Angle DAB = angle ADC = 110° (consecutive). All pairs equal → parallelogram.

Question 5: Diagonals bisect each other at O. Prove parallelogram.

Solution: AO = CO, BO = DO (given). Triangles AOB = COD (SSS). Alternate angles equal → opposite sides parallel.

Question 6: P, Q, R, S midpoints ABCD. AC diagonal. Prove SR || AC, SR = 1/2 AC.

Solution: Triangle ABC: PQ midpoint theorem → PQ || AC, PQ = 1/2 AC. Triangle ADC: SR || AC, SR = 1/2 AC.

Question 7: Prove PQRS parallelogram (from Q6).

Solution: PQ || SR (both || AC), PQ = SR = 1/2 AC → opposite sides equal/parallel.

Exercise 8.2 Solved Questions

Question 1: Parallelogram ABCD, AO/OC = BO/OD = 1/3. Prove equal areas.

Solution: Similar triangles by ratio. Areas proportional.

Question 2: Quadrilateral diagonal bisects angles. Prove parallelogram.

Solution: Angle bisector theorem on triangles.

Question 3: ABCD, AB || CD, AD || BC. Prove half diagonal property.

Solution: ASA congruence triangles ABC, ADC.

Question 4: Trapezium ABCD, AB || DC, non-parallel sides equal. Prove base angles equal.

Solution: Drop perpendiculars, congruent right triangles.

Question 5: Midpoint theorem converse proof.

Solution: If line parallel base and half, it's midsegment.

Question 6: Prove Varignon's theorem fully.

Solution: Midpoint theorem on triangles formed by diagonals.

Question 7: Application of parallelogram vector properties.

Solution: Vector AB + AD = AC diagonal.

Extra Practice Questions Solved

Extra Q1: Rhombus ABCD, angle A = 60°. Find other angles.

Solution: Opposite 60°, consecutive 120°.

Extra Q2: Rectangle ABCD, diagonal AC = 10cm. Sides?

Solution: Let length l, breadth b. l² + b² = 100.

Extra Q3: Square side 5cm. Area?

Solution: 25 cm².

Extra Q4: Kite ABCD, diagonals perpendicular. AC bisects BD. Prove angles.

Solution: Symmetry properties.

Extra Q5: Parallelogram area = base × height = 12 × 5 = 60 cm².

15 FAQs

Quadrilateral angle sum? 360 degrees.
Parallelogram opposite sides? Equal and parallel.
Consecutive angles parallelogram? Sum to 180°.
Diagonals parallelogram property? Bisect each other.
Mid-point theorem? Parallel to base, half length.
3:4:5:6 ratio angles? 36°, 48°, 90°, 120°? Wait, sum parts 18×20=360.
Rhombus all sides? Equal.
Rectangle angles? All 90°.
Trapezium has? Exactly one parallel pair.
Varignon's theorem? Midpoints form parallelogram.
Three obtuse angles possible? No.
Square diagonals? Equal, perpendicular.
Kite properties? Two pairs adjacent equal sides.
Angle sum proof? Divide into two triangles.
Vector parallelogram? Diagonal = vector sum sides.

Quadrilaterals make sense once you draw them out. Visit www.fuzymathacademy.com for my step-by-step video solutions and live practice sessions. You'll handle any quad question on your boards.