Class 9 Maths Chapter 7 Triangles – Exercise 7.4 NCERT Solutions
Introduction
Exercise 7.4 explores the exterior angle inequality property of triangles. You’ll learn how to prove that the measure of an exterior angle of a triangle is greater than either of its interior opposite angles, and apply this property to solve problems.
Key Concept
Exterior Angle Inequality:
This follows from the exterior angle property:
Since the exterior angle equals the sum, it must be greater than each individual angle.
Solved Questions (Step by Step)
Q1. In ΔABC, exterior angle at A is 120°, and interior opposite angles are ∠B = 50°, ∠C = 70°. Verify inequality.
Solution:
Exterior angle = 120°.
∠B = 50°, ∠C = 70°.
120° > 50° and 120° > 70°. Verified.
Q2. In ΔPQR, exterior angle at P is 100°, ∠Q = 40°, ∠R = 60°. Verify inequality.
Solution:
Exterior angle = 100°.
∠Q = 40°, ∠R = 60°.
100° > 40° and 100° > 60°. Verified.
Q3. In ΔXYZ, exterior angle at X is 90°, ∠Y = 30°, ∠Z = 60°. Verify inequality.
Solution:
Exterior angle = 90°.
∠Y = 30°, ∠Z = 60°.
90° > 30° and 90° > 60°. Verified.
Q4. In ΔLMN, exterior angle at L is 110°, ∠M = 50°, ∠N = 60°. Verify inequality.
Solution:
Exterior angle = 110°.
∠M = 50°, ∠N = 60°.
110° > 50° and 110° > 60°. Verified.
Q5. In ΔABC, exterior angle at B is 80°, ∠A = 35°, ∠C = 45°. Verify inequality.
Solution:
Exterior angle = 80°.
∠A = 35°, ∠C = 45°.
80° > 35° and 80° > 45°. Verified.
Q6. In ΔPQR, exterior angle at Q is 95°, ∠P = 40°, ∠R = 55°. Verify inequality.
Solution:
Exterior angle = 95°.
∠P = 40°, ∠R = 55°.
95° > 40° and 95° > 55°. Verified.
Q7. In ΔXYZ, exterior angle at Z is 70°, ∠X = 25°, ∠Y = 45°. Verify inequality.
Solution:
Exterior angle = 70°.
∠X = 25°, ∠Y = 45°.
70° > 25° and 70° > 45°. Verified.
Q8. In ΔLMN, exterior angle at M is 85°, ∠L = 50°, ∠N = 35°. Verify inequality.
Solution:
Exterior angle = 85°.
∠L = 50°, ∠N = 35°.
85° > 50° and 85° > 35°. Verified.
Q9. In ΔABC, exterior angle at C is 150°, ∠A = 70°, ∠B = 80°. Verify inequality.
Solution:
Exterior angle = 150°.
∠A = 70°, ∠B = 80°.
150° > 70° and 150° > 80°. Verified.
Q10. In ΔPQR, exterior angle at R is 120°, ∠P = 60°, ∠Q = 60°. Verify inequality.
Solution:
Exterior angle = 120°.
∠P = 60°, ∠Q = 60°.
120° > 60°. Verified.
Q11. In ΔXYZ, exterior angle at Y is 75°, ∠X = 30°, ∠Z = 45°. Verify inequality.
Solution:
Exterior angle = 75°.
∠X = 30°, ∠Z = 45°.
75° > 30° and 75° > 45°. Verified.
Q12. In ΔLMN, exterior angle at N is 135°, ∠L = 65°, ∠M = 70°. Verify inequality.
Solution:
Exterior angle = 135°.
∠L = 65°, ∠M = 70°.
135° > 65° and 135° > 70°. Verified.
Q13. In ΔABC, exterior angle at A is 105°, ∠B = 45°, ∠C = 60°. Verify inequality.
Solution:
Exterior angle = 105°.
∠B = 45°, ∠C = 60°.
105° > 45° and 105° > 60°. Verified.
Q14. In ΔPQR, exterior angle at Q is 140°, ∠P = 50°, ∠R = 90°. Verify inequality.
Solution:
Exterior angle = 140°.
∠P = 50°, ∠R = 90°.
140° > 50° and 140° > 90°. Verified.
Q15. In ΔXYZ, exterior angle at Z is 160°, ∠X = 75°, ∠Y = 85°. Verify inequality.
Solution:
Exterior angle = 160°.
∠X = 75°, ∠Y = 85°.
160° > 75° and 160° > 85°. Verified.
FAQs (10 for Exercise 7.4)
Q: What is an exterior angle of a triangle? A: An angle formed outside the triangle when one side is extended.
Q: What is the exterior angle property? A: Exterior angle = sum of two interior opposite angles.
Q: What is the exterior angle inequality? A: Exterior angle is greater than each interior opposite angle.
Q: Can an exterior angle be equal to an interior opposite angle? A: No, it is always greater.
Q: What is the maximum value of an exterior angle? A: Less than 180°.
Q: How many exterior angles can a triangle have? A: Three, one at each vertex.
Q: Why is the exterior angle property important? A: It helps solve unknown angles in triangles.
Q: What is the relation between exterior angle and adjacent interior angle? A: They form a linear pair and sum to 180°.
Q: Can the exterior angle property be extended to polygons? A: Yes, exterior angles of polygons also relate to interior angles.
Q: What is the sum of all exterior angles of a polygon? A: 360°.
Conclusion
Exercise 7.4 has 15 questions that strengthen your understanding of the exterior angle inequality property of triangles. This completes Chapter 7 of Class 9 Maths.
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