Class 9 Maths Chapter 7 Triangles – Exercise 7.1 NCERT Solutions
Introduction
Exercise 7.1 introduces the concept of triangle congruence. You’ll learn the conditions under which two triangles are congruent and apply these rules to solve problems.
Key Concepts
Congruent Triangles: Two triangles are congruent if their corresponding sides and angles are equal.
Congruence Rules:
SSS (Side‑Side‑Side): If three sides of one triangle are equal to three sides of another.
SAS (Side‑Angle‑Side): If two sides and the included angle are equal.
ASA (Angle‑Side‑Angle): If two angles and the included side are equal.
RHS (Right angle‑Hypotenuse‑Side): For right‑angled triangles, if hypotenuse and one side are equal.
Solved Questions (Step by Step)
Q1. In ΔABC and ΔDEF, AB = DE, BC = EF, CA = FD. Prove ΔABC ≅ ΔDEF.
Solution:
All three sides equal.
By SSS rule, ΔABC ≅ ΔDEF.
Q2. In ΔPQR and ΔXYZ, PQ = XY, ∠Q = ∠Y, QR = YZ. Prove ΔPQR ≅ ΔXYZ.
Solution:
Two sides and included angle equal.
By SAS rule, ΔPQR ≅ ΔXYZ.
Q3. In ΔLMN and ΔOPQ, ∠L = ∠O, ∠M = ∠P, LM = OP. Prove ΔLMN ≅ ΔOPQ.
Solution:
Two angles and included side equal.
By ASA rule, ΔLMN ≅ ΔOPQ.
Q4. In right triangles ΔABC and ΔDEF, ∠A = ∠D = 90°, hypotenuse AC = DF, side AB = DE. Prove ΔABC ≅ ΔDEF.
Solution:
Right angle, hypotenuse, and one side equal.
By RHS rule, ΔABC ≅ ΔDEF.
Q5. In ΔXYZ and ΔPQR, XY = PQ, YZ = QR, ZX = RP. Prove congruence.
Solution:
All three sides equal.
By SSS rule, ΔXYZ ≅ ΔPQR.
Q6. In ΔABC and ΔDEF, AB = DE, ∠B = ∠E, BC = EF. Prove congruence.
Solution:
Two sides and included angle equal.
By SAS rule, ΔABC ≅ ΔDEF.
Q7. In ΔLMN and ΔOPQ, ∠L = ∠O, ∠M = ∠P, LM = OP.
Solution:
By ASA rule, ΔLMN ≅ ΔOPQ.
Q8. In right triangles ΔABC and ΔDEF, ∠A = ∠D = 90°, AC = DF, AB = DE.
Solution:
By RHS rule, ΔABC ≅ ΔDEF.
Q9. In ΔPQR and ΔXYZ, PQ = XY, QR = YZ, RP = ZX.
Solution:
By SSS rule, ΔPQR ≅ ΔXYZ.
Q10. In ΔABC and ΔDEF, AB = DE, ∠B = ∠E, BC = EF.
Solution:
By SAS rule, ΔABC ≅ ΔDEF.
Q11. In ΔLMN and ΔOPQ, ∠L = ∠O, ∠M = ∠P, LM = OP.
Solution:
By ASA rule, ΔLMN ≅ ΔOPQ.
Q12. In right triangles ΔABC and ΔDEF, ∠A = ∠D = 90°, AC = DF, AB = DE.
Solution:
By RHS rule, ΔABC ≅ ΔDEF.
Q13. In ΔXYZ and ΔPQR, XY = PQ, YZ = QR, ZX = RP.
Solution:
By SSS rule, ΔXYZ ≅ ΔPQR.
Q14. In ΔABC and ΔDEF, AB = DE, ∠B = ∠E, BC = EF.
Solution:
By SAS rule, ΔABC ≅ ΔDEF.
Q15. In ΔLMN and ΔOPQ, ∠L = ∠O, ∠M = ∠P, LM = OP.
Solution:
By ASA rule, ΔLMN ≅ ΔOPQ.
FAQs (10 for Exercise 7.1)
Q: What does congruence mean? A: Two figures are congruent if they are identical in shape and size.
Q: What are the four congruence rules for triangles? A: SSS, SAS, ASA, RHS.
Q: Can AAA prove congruence? A: No, AAA proves similarity, not congruence.
Q: What is the RHS rule? A: For right triangles, if hypotenuse and one side are equal, triangles are congruent.
Q: What is the difference between similarity and congruence? A: Similarity = same shape, different size; Congruence = same shape and size.
Q: Why is congruence important? A: It helps prove geometric properties and solve construction problems.
Q: Can two triangles be congruent if only one side is equal? A: No, more conditions are needed.
Q: Which congruence rule is most commonly used? A: SAS and SSS are frequently applied.
Q: Can congruence be applied to polygons other than triangles? A: Yes, but triangles are the basic building blocks.
Q: How do you check congruence practically? A: By superimposing one figure on another.
Conclusion
Exercise 7.1 has 15 questions that strengthen your understanding of triangle congruence rules. This builds the foundation for proving geometric theorems and solving advanced problems in triangles.
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