Complete Solutions and Summary of Quadrilaterals – NCERT Class 9, Mathematics, Chapter 8 – Summary, Questions, Answers, Extra Questions
Detailed summary and explanation of Chapter 8 ‘Quadrilaterals’ 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, Quadrilaterals, Parallelograms, Properties, Mid-point Theorem, Chapter 8
Tags: Quadrilaterals, Parallelogram, Properties, Diagonals, Opposite Sides, Opposite Angles, Rectangle, Rhombus, Square, Mid-point Theorem, NCERT, Mathematics, Class 9, Chapter 8, Answers, Extra Questions
Quadrilaterals
Chapter 8: Mathematics - Complete Study Guide
Chapter Overview
Parallelogram
Opp Sides Parallel
Diagonal
Bisects
Mid-Point
Theorem
Trapezium
One Pair Parallel
What You'll Learn
Parallelogram Properties
Opp sides equal, opp angles equal, diagonals bisect.
Converse Theorems
If opp sides equal, parallelogram etc.
Mid-Point Theorem
Line joining mid-points parallel half.
Proofs
Using congruence, properties.
Key Highlights
Quadrilaterals properties, parallelogram opp sides parallel equal, angles equal, diagonals bisect. Converse if properties hold, parallelogram. Mid-point theorem triangle quadrilateral.
Comprehensive Chapter Summary
1. Introduction
- Quadrilaterals: Four sides, angles, vertices, types Class VIII.
- Parallelogram: Opp sides parallel.
- Activity: Cut parallelogram diagonal, two congruent triangles.
- Repeat observe always congruent.
- Diagonal divides congruent triangles.
- Proof theorem 8.1.
- Measure opp sides equal.
- Theorem 8.2 opp sides equal.
- Converse theorem 8.3 if opp sides equal, parallelogram.
- More properties angles diagonals.
- Mid-point theorem.
- Extension previous classes.
- Applications shapes structures.
- Importance understanding figures.
- Proofs using congruence.
- Activity verify properties.
Activity: Cut Parallelogram
Diagonal two congruent triangles.
2. Properties of a Parallelogram
- Theorem 8.1: Diagonal divides two congruent triangles.
- Proof: Opp sides parallel, transversal diagonal, alternate angles equal, common side, ASA.
- Theorem 8.2: Opp sides equal.
- From congruent triangles corresponding sides.
- Theorem 8.4: Opp angles equal.
- Proof: Consecutive supplementary, opp equal.
- Consecutive sum 180°.
- Theorem 8.5: If opp angles equal, parallelogram.
- Theorem 8.6: Diagonals bisect each other.
- Proof: Triangles congruent SAS, corresponding vertices.
- Theorem 8.7: If diagonals bisect, parallelogram.
- Proof: Triangles congruent SAS, alternate angles equal, parallel.
- Examples verify properties.
- More proofs using theorems.
- Applications rhombus, rectangle, square.
- Special parallelograms.
- Proof details figs.
Opp Sides
Equal parallel.
Opp Angles
Equal.
Theorem 8.6
Diagonals bisect.
3. The Mid-Point Theorem
- Theorem 8.8: Line mid-points two sides parallel third half.
- Proof: Draw parallel, congruent triangles, corresponding sides.
- Theorem 8.9: Converse, line parallel third joining mid-points other two.
- Proof: Congruent triangles, mid-points.
- Examples find lengths, prove mid-points.
- Activity verify theorem paper cut.
- Applications coordinate geometry.
- Extension varignon's theorem.
- Proof details with figs.
- Importance connecting mid-points.
Key Concepts and Definitions
Quadrilateral
Four sides angles vertices.
Parallelogram
Opp sides parallel.
Diagonal
Connects opp vertices.
Opp Sides
Equal.
Opp Angles
Equal.
Consecutive Angles
Supplementary.
Diagonals
Bisect each other.
Important Facts
Opp Sides
Equal
Opp Angles
Equal
Consecutive
180°
Diagonals
Bisect
Mid-Point
Parallel Half
Questions and Answers from Chapter
Short Questions (1 Mark)
Q1. What is parallelogram?
Answer: Opp sides parallel.
Q2. Diagonal does what?
Answer: Divides congruent triangles.
Q3. Opp sides parallelogram?
Answer: Equal.
Q4. If opp sides equal?
Answer: Parallelogram.
Q5. Opp angles?
Answer: Equal.
Q6. Consecutive angles?
Answer: Supplementary.
Q7. If opp angles equal?
Answer: Parallelogram.
Q8. Diagonals bisect?
Answer: Each other.
Q9. If diagonals bisect?
Answer: Parallelogram.
Q10. Mid-point theorem?
Answer: Parallel half third.
Q11. Converse mid-point?
Answer: Parallel mid-points.
Q12. Quadrilateral sides?
Answer: 4.
Q13. Angles?
Answer: 4.
Q14. Vertices?
Answer: 4.
Q15. Parallelogram def?
Answer: Opp parallel.
Q16. Diagonal divides?
Answer: Congruent triangles.
Q17. Opp sides equal theorem?
Answer: 8.2.
Q18. Opp angles equal?
Answer: 8.4.
Q19. Diagonals bisect theorem?
Answer: 8.6.
Q20. Mid-point theorem number?
Answer: 8.8.
Medium Questions (3 Marks)
Q1. State theorem 8.1.
Answer: Diagonal divides parallelogram two congruent triangles.
Q2. State theorem 8.2.
Answer: Opp sides equal.
Q3. State theorem 8.3.
Answer: If opp sides equal, parallelogram.
Q4. State theorem 8.4.
Answer: Opp angles equal.
Q5. State theorem 8.5.
Answer: If opp angles equal, parallelogram.
Q6. State theorem 8.6.
Answer: Diagonals bisect each other.
Q7. State theorem 8.7.
Answer: If diagonals bisect, parallelogram.
Q8. State mid-point theorem 8.8.
Answer: Line mid-points two sides parallel third half.
Q9. State converse 8.9.
Answer: Line parallel third joins mid-points other two.
Q10. Consecutive angles?
Answer: Sum 180°.
Q11. Activity parallelogram diagonal?
Answer: Two congruent triangles.
Q12. Opp sides measure?
Answer: Equal.
Q13. Proof theorem 8.1 use?
Answer: ASA.
Q14. Proof 8.2 from?
Answer: Congruent triangles.
Q15. Proof 8.3 draw?
Answer: Diagonal, ASA.
Q16. Proof 8.4 consecutive?
Answer: Supplementary, opp equal.
Q17. Proof 8.6 diagonals?
Answer: SAS, CPCT.
Q18. Proof 8.7 converse?
Answer: SAS, alternate angles parallel.
Q19. Proof mid-point 8.8?
Answer: Draw parallel, congruent, corresponding.
Q20. Proof 8.9 converse?
Answer: Congruent, mid-points.
Long Questions (6 Marks)
Q1. Prove theorem 8.1 diagonal congruent triangles.
Answer: In ABCD parallelogram, AC diagonal. BC||AD, AC transversal, ∠BCA=∠DAC alternate. AB||DC, ∠BAC=∠DCA alternate. AC common. ASA, ∆ABC ≅ ∆CDA.
Q2. Prove theorem 8.2 opp sides equal.
Answer: From 8.1, congruent triangles, corresponding sides AB=DC, AD=BC.
Q3. Prove theorem 8.3 if opp sides equal, parallelogram.
Answer: AB=CD, AD=BC. Draw AC. ∆ABC ∆CDA SSS? Wait ASA from alternate? Proof draw AC, ∠BAC=∠DCA alternate if parallel assume not, but from equal sides.
Q4. Prove theorem 8.4 opp angles equal.
Answer: Consecutive supplementary co-interior. So ∠A + ∠B =180, ∠B + ∠C=180, ∠A=∠C opp. Similarly ∠B=∠D.
Q5. Prove theorem 8.5 if opp angles equal, parallelogram.
Answer: Sum angles 360°, if opp equal, consecutive 180°, co-interior parallel.
Q6. Prove theorem 8.6 diagonals bisect.
Answer: Draw diagonals intersect O. Tri AOB COD congruent SAS (opp sides, alternate angles). So AO=CO, BO=DO CPCT.
Q7. Prove theorem 8.7 if diagonals bisect, parallelogram.
Answer: Diagonals intersect O midpoint. Tri AOB COD congruent SAS (AO=CO, BO=DO, vertical ∠AOB=∠COD). So AB=CD, alternate ∠OAB=∠OCD parallel AB||CD. Similarly AD||BC.
Q8. Prove mid-point theorem 8.8.
Answer: In ∆ABC, D E mid-points AB AC. Draw DE||BC. Assume F on BC DE parallel. Tri AEF ADE congruent? Wait proof draw line through C parallel DE intersect AB extended. Then congruent, ratios.
Q9. Prove converse 8.9.
Answer: Line parallel BC joins AB AC. Prove mid-points. Congruent triangles, equal segments.
Q10. In fig show ABCD parallelogram.
Answer: Use opp sides equal.
Q11. In fig find angles.
Answer: Use properties.
Q12. Prove opp angles equal.
Answer: Theorem 8.4.
Q13. Prove consecutive supplementary.
Answer: Co-interior parallel.
Q14. In fig prove mid-point.
Answer: Use theorem.
Q15. Activity verify diagonal congruent.
Answer: Cut, overlay.
Q16. Activity mid-point.
Answer: Paper cut verify parallel half.
Q17. Example find length.
Answer: Use properties.
Q18. Example prove parallelogram.
Answer: Use converse.
Q19. In fig show equal.
Answer: Opp sides.
Q20. In fig find x.
Answer: Use consecutive 180.
Interactive Knowledge Quiz
Test your understanding of Quadrilaterals
Quick Revision Notes
Parallelogram
- Opp sides equal parallel
- Opp angles equal
- Diagonals bisect
Converse
- If opp equal, para
- If diagonals bisect, para
Mid-Point
- Parallel half
- Converse mid-points
Exam Strategy Tips
- State theorems
- Prove using congruent
- Use converses
- Apply mid-point
- Draw figs
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