Class 9 Maths Chapter 7 Triangles – Exercise 7.3 NCERT Solutions
Introduction
Exercise 7.3 explores the properties of isosceles triangles. You’ll learn how to prove that angles opposite equal sides are equal, and conversely, sides opposite equal angles are equal. These properties are fundamental in geometry and proofs.
Key Concepts
Isosceles Triangle: A triangle with two equal sides.
Property 1: Angles opposite equal sides are equal.
Property 2: Sides opposite equal angles are equal.
Solved Questions (Step by Step)
Q1. In ΔABC, AB = AC. Prove ∠B = ∠C.
Solution:
By property of isosceles triangle, angles opposite equal sides are equal.
Hence, ∠B = ∠C.
Q2. In ΔPQR, PQ = PR. Prove ∠Q = ∠R.
Solution:
Angles opposite equal sides are equal.
Hence, ∠Q = ∠R.
Q3. In ΔXYZ, ∠Y = ∠Z. Prove XY = XZ.
Solution:
Sides opposite equal angles are equal.
Hence, XY = XZ.
Q4. In ΔLMN, LM = LN. Prove ∠M = ∠N.
Solution:
Angles opposite equal sides are equal.
Hence, ∠M = ∠N.
Q5. In ΔABC, ∠B = ∠C. Prove AB = AC.
Solution:
Sides opposite equal angles are equal.
Hence, AB = AC.
Q6. In ΔPQR, PQ = PR. Find ∠Q and ∠R if ∠P = 40°.
Solution:
∠Q = ∠R.
∠Q + ∠R = 140°.
∠Q = ∠R = 70°.
Q7. In ΔXYZ, XY = XZ. Find ∠Y and ∠Z if ∠X = 50°.
Solution:
∠Y = ∠Z.
∠Y + ∠Z = 130°.
∠Y = ∠Z = 65°.
Q8. In ΔLMN, LM = LN. Find ∠M and ∠N if ∠L = 30°.
Solution:
∠M = ∠N.
∠M + ∠N = 150°.
∠M = ∠N = 75°.
Q9. In ΔABC, AB = AC. Find ∠B and ∠C if ∠A = 100°.
Solution:
∠B = ∠C.
∠B + ∠C = 80°.
∠B = ∠C = 40°.
Q10. In ΔPQR, PQ = PR. Find ∠Q and ∠R if ∠P = 90°.
Solution:
∠Q = ∠R.
∠Q + ∠R = 90°.
∠Q = ∠R = 45°.
Q11. In ΔXYZ, XY = XZ. Find ∠Y and ∠Z if ∠X = 120°.
Solution:
∠Y = ∠Z.
∠Y + ∠Z = 60°.
∠Y = ∠Z = 30°.
Q12. In ΔLMN, LM = LN. Find ∠M and ∠N if ∠L = 80°.
Solution:
∠M = ∠N.
∠M + ∠N = 100°.
∠M = ∠N = 50°.
Q13. In ΔABC, ∠B = ∠C. Find AB and AC if AB = 5 cm.
Solution:
AB = AC.
Hence, AC = 5 cm.
Q14. In ΔPQR, ∠Q = ∠R. Find PQ and PR if PQ = 7 cm.
Solution:
PQ = PR.
Hence, PR = 7 cm.
Q15. In ΔXYZ, ∠Y = ∠Z. Find XY and XZ if XY = 10 cm.
Solution:
XY = XZ.
Hence, XZ = 10 cm.
FAQs (10 for Exercise 7.3)
Q: What is an isosceles triangle? A: A triangle with two equal sides.
Q: What is the property of angles in an isosceles triangle? A: Angles opposite equal sides are equal.
Q: What is the property of sides in an isosceles triangle? A: Sides opposite equal angles are equal.
Q: Can an isosceles triangle be equilateral? A: Yes, if all three sides are equal.
Q: Can an isosceles triangle be right‑angled? A: Yes, one angle can be 90°.
Q: What is the sum of angles in a triangle? A: 180°.
Q: How many equal angles does an isosceles triangle have? A: Two equal angles.
Q: How many equal sides does an isosceles triangle have? A: Two equal sides.
Q: Why are isosceles triangle properties important? A: They simplify proofs and constructions.
Q: What is the converse property of isosceles triangles? A: If two angles are equal, their opposite sides are equal.
Conclusion
Exercise 7.3 has 15 questions that strengthen your understanding of isosceles triangle properties. This builds the foundation for proving geometric theorems and solving advanced problems in triangles.
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