Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry – Exercise 5.2 NCERT Solutions

Introduction

Exercise 5.2 focuses on Euclid’s axioms and postulates and their applications in proving simple geometric results. Students learn how to apply logical reasoning to establish relationships between lines, angles, and points. This exercise builds the foundation for deductive geometry.

Key Concepts

1. Euclid’s Axioms:

   • Things equal to the same thing are equal to one another.

   • If equals are added to equals, the wholes are equal.

   • If equals are subtracted from equals, the remainders are equal.

2. Euclid’s Postulates:

   • A straight line may be drawn from any point to any other point.

   • A terminated line can be extended indefinitely.

   • A circle can be drawn with any centre and radius.

   • All right angles are equal to one another.

Common Mistakes

• Confusing axioms with postulates.

• Skipping logical steps in proofs.

• Misinterpreting “things equal to the same thing” axiom.

• Forgetting that postulates are assumptions accepted without proof.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. State Euclid’s first axiom with example.

$$\text{<span style="font-family: verdana;"><b>If </b></span>}\mathrm{<span\; style="font-family:\; verdana;"><b>a</b></span>}<span\; style="font-family:\; verdana;"><b>=</b></span>\mathrm{<span\; style="font-family:\; verdana;"><b>b</b></span>}\text{<span style="font-family: verdana;"><b> and </b></span>}\mathrm{<span\; style="font-family:\; verdana;"><b>b</b></span>}<span\; style="font-family:\; verdana;"><b>=</b></span>\mathrm{<span\; style="font-family:\; verdana;"><b>c</b></span>}<span\; style="font-family:\; verdana;"><b>,</b></span>\text{<span style="font-family: verdana;"><b> then </b></span>}\mathrm{<span\; style="font-family:\; verdana;"><b>a</b></span>}<span\; style="font-family:\; verdana;"><b>=</b></span>\mathrm{<span\; style="font-family:\; verdana;"><b>c</b></span>}\mathrm{<span\; style="font-family:\; verdana;"><b>.</b></span>}$$

Example: If two sides of a triangle are equal to a third side, they are equal to each other.

Q2. State Euclid’s second axiom with example.

$$\text{<span style="font-family: verdana;"><b>If equals are added to equals, the wholes are equal.</b></span>}$$

Example: If $a=b$, then $a+c=b+c$.

Q3. State Euclid’s third axiom with example.

$$\text{<span style="font-family: verdana;"><b>If equals are subtracted from equals, the remainders are equal.</b></span>}$$

Example: If $a=b$, then $a-c=b-c$.

Q4. State Euclid’s fourth axiom. Things which coincide with one another are equal to one another.

Q5. State Euclid’s fifth axiom. The whole is greater than the part.

Q6. State Euclid’s first postulate. A straight line may be drawn from any point to any other point.

Q7. State Euclid’s second postulate. A terminated line can be extended indefinitely.

Q8. State Euclid’s third postulate. A circle can be drawn with any centre and radius.

Q9. State Euclid’s fourth postulate. All right angles are equal to one another.

Q10. State Euclid’s fifth postulate. If a straight line falling on two straight lines makes interior angles on the same side less than two right angles, then the two lines, if produced indefinitely, meet on that side.

FAQs (10)

FAQ1. What is difference between axiom and postulate? Axiom applies universally, postulate applies to geometry.

FAQ2. What is Euclid’s first axiom? Things equal to the same thing are equal to one another.

FAQ3. What is Euclid’s fifth axiom? The whole is greater than the part.

FAQ4. What is Euclid’s first postulate? A straight line may be drawn from any point to any other point.

FAQ5. What is Euclid’s second postulate? A terminated line can be extended indefinitely.

FAQ6. What is Euclid’s third postulate? A circle can be drawn with any centre and radius.

FAQ7. What is Euclid’s fourth postulate? All right angles are equal to one another.

FAQ8. What is Euclid’s fifth postulate? It describes parallel lines meeting when angles are less than two right angles.

FAQ9. Why are Euclid’s axioms important? They form the logical foundation of geometry.

FAQ10. Why is Exercise 5.2 important? It develops deductive reasoning in geometry.

Conclusion

Exercise 5.2 covers Euclid’s axioms and postulates with solved examples and FAQs. Mastering these concepts helps students understand the logical structure of geometry.

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