Class 9 Maths Chapter 6 Lines and Angles – Exercise 6.3 NCERT Solutions

Introduction

Exercise 6.3 explores the angle sum property of triangles. You’ll learn how to prove that the sum of the angles of a triangle is 180°, and apply this property to solve problems involving missing angles.

Key Concept

• Angle Sum Property of Triangle:

$$\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C={180}^{\circ }$$

• Exterior Angle Property: Exterior angle = sum of two interior opposite angles.

Solved Questions (Step by Step)

Q1. In a triangle, two angles are 50° and 60°. Find the third angle.

Solution:

$$\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C={180}^{\circ }$$

$${50}^{\circ }+{60}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={70}^{\circ }$$

Q2. In a triangle, two angles are 40° and 75°. Find the third angle.

Solution:

$${40}^{\circ }+{75}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={65}^{\circ }$$

Q3. In a triangle, two angles are 90° and 45°. Find the third angle.

Solution:

$${90}^{\circ }+{45}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={45}^{\circ }$$

Q4. In a triangle, two angles are 80° and 55°. Find the third angle.

Solution:

$${80}^{\circ }+{55}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={45}^{\circ }$$

Q5. In a triangle, two angles are 30° and 60°. Find the third angle.

Solution:

$${30}^{\circ }+{60}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={90}^{\circ }$$

Q6. In a triangle, two angles are 100° and 40°. Find the third angle.

Solution:

$${100}^{\circ }+{40}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={40}^{\circ }$$

Q7. In a triangle, two angles are 20° and 70°. Find the third angle.

Solution:

$${20}^{\circ }+{70}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={90}^{\circ }$$

Q8. In a triangle, two angles are 85° and 65°. Find the third angle.

Solution:

$${85}^{\circ }+{65}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={30}^{\circ }$$

Q9. In a triangle, two angles are 95° and 45°. Find the third angle.

Solution:

$${95}^{\circ }+{45}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={40}^{\circ }$$

Q10. In a triangle, two angles are 70° and 50°. Find the third angle.

Solution:

$${70}^{\circ }+{50}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={60}^{\circ }$$

Q11. In a triangle, two angles are 120° and 30°. Find the third angle.

Solution:

$${120}^{\circ }+{30}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={30}^{\circ }$$

Q12. In a triangle, two angles are 75° and 75°. Find the third angle.

Solution:

$${75}^{\circ }+{75}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={30}^{\circ }$$

Q13. In a triangle, two angles are 60° and 60°. Find the third angle.

Solution:

$${60}^{\circ }+{60}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={60}^{\circ }$$

Q14. In a triangle, two angles are 45° and 45°. Find the third angle.

Solution:

$${45}^{\circ }+{45}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={90}^{\circ }$$

Q15. In a triangle, two angles are 30° and 30°. Find the third angle.

Solution:

$${30}^{\circ }+{30}^{\circ }+\mathrm{\angle }C={180}^{\circ }⟹\mathrm{\angle }C={120}^{\circ }$$

FAQs (10 for Exercise 6.3)

1. Q: What is the angle sum property of a triangle? A: The sum of the three angles of a triangle is 180°.

2. Q: What is an exterior angle of a triangle? A: Angle formed outside the triangle when one side is extended.

3. Q: What is the exterior angle property? A: Exterior angle = sum of two interior opposite angles.

4. Q: Can a triangle have two right angles? A: No, because sum would exceed 180°.

5. Q: Can a triangle have two obtuse angles? A: No, because sum would exceed 180°.

6. Q: What is an equilateral triangle? A: A triangle with all angles equal to 60°.

7. Q: What is an isosceles triangle? A: A triangle with two equal sides and two equal angles.

8. Q: What is a right‑angled triangle? A: A triangle with one angle equal to 90°.

9. Q: Why is the angle sum property important? A: It helps solve missing angles and proves geometric facts.

10. Q: Can the angle sum property be extended to polygons? A: Yes, sum of interior angles of an n‑sided polygon = $(n-2)\times {180}^{\circ }$.

Conclusion

Exercise 6.3 has 15 questions that strengthen your understanding of the angle sum property of triangles and related proofs. This builds the foundation for studying polygons and advanced geometry.

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