Complete Solutions and Summary of Triangles – NCERT Class 9, Mathematics, Chapter 7 – Summary, Questions, Answers, Extra Questions
Detailed summary and explanation of Chapter 7 ‘Triangles’ with all question answers, extra questions, and solutions from NCERT Class IX, Mathematics.
Updated: 3 weeks ago
Categories: NCERT, Class IX, Mathematics, Summary, Extra Questions, Triangles, Congruence, Properties of Triangles, Chapter 7
Tags: Triangles, Congruence, SAS, ASA, AAS, SSS, RHS, Isosceles Triangle, Equilateral Triangle, Angle Properties, Side Properties, NCERT, Class 9, Mathematics, Chapter 7, Answers, Extra Questions
Triangles
Chapter 7: Mathematics - Complete Study Guide
Chapter Overview
SSS
Side-Side-Side
SAS
Side-Angle-Side
ASA
Angle-Side-Angle
RHS
Right-Hyp-Side
What You'll Learn
Triangle Congruence
Criteria SAS, ASA, SSS, RHS.
Properties
Angles opposite equal sides equal.
Inequalities
Sum sides > third, difference < third.
Proofs
Using congruence.
Key Highlights
Triangles congruent if sides, angles match criteria. Properties isosceles, inequalities relations sides angles. Prove using CPCT.
Comprehensive Chapter Summary
1. Introduction
- Triangles: Three sides, angles, vertices, e.g., ∆ABC.
- Properties studied earlier.
- Congruence, rules, more properties, inequalities.
- Closed figure three intersecting lines.
- Denoted ∆ABC.
- Sides AB, BC, CA; angles ∠A, ∠B, ∠C; vertices A,B,C.
- Chapter details congruence rules, proofs.
- Verified activities earlier.
- Extension previous knowledge.
- Importance understanding shapes.
- Applications engineering, architecture.
- Basic triangle facts recall.
- Sum angles 180°.
- Types: equilateral, isosceles, scalene.
- Acute, obtuse, right.
- Exterior angle = opposite interiors sum.
Example: Triangle
∆ABC fig.
2. Congruence of Triangles
- Congruent figures: Same shape size, e.g., photos, bangles, coins.
- Circles same radius congruent.
- Squares same side congruent.
- Equilateral triangles same side congruent.
- Ice tray moulds congruent.
- Refills same size congruent.
- Triangles congruent if sides angles equal corresponding.
- CPCT: Corresponding parts congruent triangles.
- Order vertices important correspondence.
- Superimpose check congruence.
- Not congruent if not overlay completely.
- Examples congruent non.
- More examples daily life.
- Importance manufacturing identical parts.
- Notation ≅ for congruence.
- Corresponding sides opposite equal angles.
- Proofs use congruence.
Congruent
Same shape size.
CPCT
Corresponding parts.
Example: Congruent Figures
Photos, coins.
3. Criteria for Congruence of Triangles
- SAS: Two sides included angle equal.
- ASA: Two angles included side equal.
- AAS: Two angles non-included side equal.
- SSS: Three sides equal.
- RHS: Right angle, hypotenuse, side equal.
- Not AAA, SSA generally.
- Examples check congruence criteria.
- Proofs using criteria.
- ASA = AAS equivalent.
- Activity verify SAS.
- More activities ASA, SSS, RHS.
- Ambiguous SSA case.
- Importance correct criteria.
- Applications surveying, construction.
- CPCT after congruence.
Axiom SAS
Two triangles congruent if two sides included angle equal corresponding.
Theorem ASA
Two angles included side.
4. Some Properties of a Triangle
- Isosceles: Angles opposite equal sides equal.
- Proof using congruence.
- Converse: Sides opposite equal angles equal.
- Examples find angles sides.
- More proofs using properties.
- Base angles isosceles equal.
- Equilateral all equal.
- Applications symmetry.
- Extension equilateral 60°.
- Proof details with fig.
5. Inequalities in a Triangle
- Sum two sides > third.
- Difference two sides < third.
- Side opposite greater angle longer.
- Converse: Greater side opposite greater angle.
- Proofs using contradiction, congruence.
- Examples check triangle possible, compare sides angles.
- Applications stability structures.
- Inequality theorem.
- Strict inequalities sides angles.
- Equality equilateral.
Key Concepts and Definitions
Triangle
Three sides, angles, vertices.
Congruent
Same shape size.
SAS
Side angle side.
ASA
Angle side angle.
SSS
Side side side.
RHS
Right hyp side.
Isosceles
Equal sides angles.
Important Facts
SAS
Congruence
ASA
Congruence
SSS
Congruence
RHS
Right
Inequality
Sum > Third
Questions and Answers from Chapter
Short Questions (1 Mark)
Q1. What is triangle?
Answer: Three lines intersecting.
Q2. Number sides triangle?
Answer: 3.
Q3. Number angles?
Answer: 3.
Q4. Number vertices?
Answer: 3.
Q5. Congruent means?
Answer: Same shape size.
Q6. CPCT?
Answer: Corresponding parts.
Q7. SAS?
Answer: Side angle side.
Q8. ASA?
Answer: Angle side angle.
Q9. SSS?
Answer: Side side side.
Q10. RHS?
Answer: Right hyp side.
Q11. Isosceles property?
Answer: Angles opp equal sides equal.
Q12. Inequality sum?
Answer: Two sides > third.
Q13. Inequality diff?
Answer: Two sides > diff.
Q14. Greater angle opp?
Answer: Longer side.
Q15. Greater side opp?
Answer: Greater angle.
Q16. AAA congruence?
Answer: No.
Q17. SSA congruence?
Answer: No.
Q18. RHS for?
Answer: Right triangles.
Q19. Isosceles base angles?
Answer: Equal.
Q20. Triangle angle sum?
Answer: 180°.
Medium Questions (3 Marks)
Q1. Define congruent triangles.
Answer: Same sides angles corresponding.
Q2. State SAS congruence.
Answer: Two sides included angle equal.
Q3. State ASA.
Answer: Two angles included side.
Q4. State SSS.
Answer: Three sides equal.
Q5. State RHS.
Answer: Right angle, hypotenuse, side.
Q6. Why not AAA?
Answer: Similar not congruent.
Q7. Why not SSA?
Answer: Ambiguous case.
Q8. Isosceles theorem.
Answer: Angles opp equal sides equal.
Q9. Converse isosceles.
Answer: Sides opp equal angles equal.
Q10. Inequality theorem 1.
Answer: Sum > third.
Q11. Inequality 2.
Answer: Diff < third.
Q12. Greater side opp.
Answer: Greater angle.
Q13. Greater angle opp.
Answer: Longer side.
Q14. Check 5,7,12 triangle?
Answer: Yes, 5+7>12 etc.
Q15. Check 8,6,2?
Answer: No, 6+2=8 not >.
Q16. In triangle, angle 70°, sides?
Answer: Opp 70° longest if obtuse? Wait standard.
Q17. SAS example.
Answer: Two sides angle between.
Q18. RHS for right triangles.
Answer: Yes.
Q19. CPCT use after?
Answer: Congruence established.
Q20. Order vertices important?
Answer: For correspondence.
Long Questions (6 Marks)
Q1. In fig if AB=AC, prove angle B=angle C.
Answer: Draw bisector AD. Tri ABD ACD congruent SAS. Angle B=angle C CPCT.
Q2. In fig AB=AC, angle B=angle C prove.
Answer: Assume not, contradiction.
Q3. Line l points A B C, AB=BC prove angle opp equal.
Answer: Use isosceles.
Q4. Prove sum two sides > third.
Answer: Extend side, prove inequality.
Q5. Prove greater angle opp longer side.
Answer: Assume not, construct equal, contradiction.
Q6. Prove greater side opp greater angle.
Answer: Converse, similar.
Q7. In right triangle, hypotenuse longest prove.
Answer: Opp 90° greatest.
Q8. Show sum other two > third.
Answer: Inequality theorem.
Q9. Show diff < third.
Answer: From sum.
Q10. In fig show angle B > angle A.
Answer: Side opp B longer.
Q11. In fig show side BC > AB.
Answer: Opp larger angle.
Q12. In fig show angle A + angle B = angle A + angle C.
Answer: Use isosceles.
Q13. In fig show angle B = angle C.
Answer: Isosceles.
Q14. Prove SAS congruence.
Answer: Axiom.
Q15. Prove ASA.
Answer: Using SAS.
Q16. Prove SSS.
Answer: Construct, SAS.
Q17. Prove RHS.
Answer: SSS with constructed.
Q18. Show congruent if SAS.
Answer: Criteria.
Q19. In fig show congruent ASA.
Answer: Common side, equal angles.
Q20. In fig show congruent SSS.
Answer: Three sides equal.
Interactive Knowledge Quiz
Test your understanding of Triangles
Quick Revision Notes
Congruence
- SAS
- ASA
- SSS
- RHS
Properties
- Isosceles angles
- Sum 180°
Inequalities
- Sum > third
- Diff < third
- Greater angle longer side
Exam Strategy Tips
- Draw figures
- Use criteria
- Prove properties
- Check inequalities
- CPCT use
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