Class 9 Maths Chapter 5 Introduction to Euclid's Geometry NCERT Solutions: Exercise 5.1 + Extra Questions Solved

Introduction

Chapter 5 introduces Euclid's geometry from his famous book "Elements." He used definitions, axioms (common truths), and postulates (geometry rules) as the foundation for all proofs. No calculations here, just logic.

This builds your reasoning skills. One exercise in NCERT with true/false plus proofs. I've added extra questions to practice more.

Key Concepts

No formulas, but these basics:

• Point: no part, no size.

• Line: length without breadth.

• Surface: no thickness.

• Axioms: Things equal to same thing are equal; if equals added to equals, wholes equal; whole greater than part.

• Postulates: Unique line through two points; lines extend forever; circles from any center/radius; all right angles equal; one parallel through point outside line.

Exercise 5.1 Solved Questions

Question 1: True or False (reasons):

(i) Only one straight line through a single point. False. Needs two points (Postulate 1). One point allows infinite lines.

(ii) Two distinct points, infinite lines possible. False. Exactly one line (Postulate 1).

(iii) A terminated line can be produced indefinitely. True. Postulate 2.

(iv) Two circles equal only if radii equal. True. Superimpose (Axiom 4); if they coincide everywhere, radii match.

(v) If AB = PQ and PQ = XY, then AB = XY. True. Axiom 1 (equals to same are equal).

Question 2: Hoyt's verifier shows equal segments on parallel lines cut by transversal. True by congruence or axioms.

Question 3: If two circular patches grow with same radius, areas equal. True. Radii equal by superposition.

Question 4: If C between A and B, AC = CB. Show AC = 1/2 AB.

Solution: Given AC = CB. Add AC to both sides: AC + AC = CB + AC. So 2 AC = AB (Axiom 2). Thus AC = (1/2) AB.

Question 5: How many lines through two distinct points? One (Postulate 1).

Extra Practice Questions Solved

Extra Q1: If two triangles are congruent, their areas equal. Reason? True. Superposition (Axiom 4) makes them coincide.

Extra Q2: Prove: If a > b and b > c, then a > c. Axiom 5 (whole > part, transitivity implied).

Extra Q3: Circle with center O radius 5 cm. Can I draw another identical? Yes. Postulate 3, same center and radius.

Extra Q4: Points A, B, C collinear, B between A,C. If AB = 3 cm, BC = 3 cm, find AC. 6 cm. AB + BC = AC (Axiom 2).

Extra Q5: Right angles at P and Q equal? Yes. Postulate 4.

Extra Q6: Line L, point P not on L. How many parallels through P? One (Postulate 5).

Extra Q7: If XY = YZ, and YZ = AB, then XY = AB. Axiom 1.

15 FAQs

1. What is Euclid's main book? Elements.

2. Point has what properties? No parts.

3. Lines through one point? Infinite possible.

4. Lines through two points? Exactly one.

5. Postulate 2 says? Lines extend indefinitely.

6. Draw circle needs? Center and radius.

7. Postulate 4? Right angles equal.

8. Axiom 5? Whole greater than part.

9. Superposition is? Axiom 4.

10. Equal circles mean? Equal radii.

11. C midpoint AB? AC = 1/2 AB.

12. Axioms different from postulates? Axioms general truths.

13. Fifth postulate? Unique parallel line.

14. Infinite lines two points? No, only one.

15. If equals added? Results equal (Axiom 2).

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