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

Introduction

Exercise 5.2 applies Euclid’s axioms and postulates to prove basic geometric facts. You’ll learn how to use logical reasoning to establish truths, such as equality of segments, angles, and properties of figures.

Key Concepts

• Axioms: Universal truths (e.g., things equal to the same thing are equal to one another).

• Postulates: Assumptions specific to geometry (e.g., a straight line can be drawn joining any two points).

• Logical Proofs: Step‑by‑step reasoning based on axioms and postulates.

Solved Questions (Step by Step)

Q1. Show that things equal to the same thing are equal to one another.

Solution:

• Let $a=c$ and $b=c$.

• Then $a=b$.

• This proves Euclid’s first axiom.

Q2. If equals are added to equals, prove wholes are equal.

Solution:

• Let $a=b$. Add $c$ to both: $a+c=b+c$.

• Hence, wholes are equal.

Q3. If equals are subtracted from equals, prove remainders are equal.

Solution:

• Let $a=b$. Subtract $c$: $a-c=b-c$.

• Hence, remainders are equal.

Q4. Prove that all right angles are equal.

Solution:

• By Euclid’s fourth postulate, all right angles are equal to one another.

Q5. Prove that the whole is greater than the part.

Solution:

• If $a=b+c$, then clearly $a>b$.

• Hence, whole is greater than part.

Q6. Show that if two lines intersect, they cannot be parallel.

Solution:

• Parallel lines never meet.

• Intersecting lines meet at a point.

• Hence, they cannot be parallel.

Q7. Prove that two distinct lines cannot have more than one point in common.

Solution:

• If they had two points in common, they would coincide.

• Hence, only one point of intersection is possible.

Q8. Prove that a terminated line can be produced indefinitely.

Solution:

• By Euclid’s second postulate, a line segment can be extended infinitely.

Q9. Prove that a circle can be drawn with any centre and radius.

Solution:

• By Euclid’s third postulate, a circle can be described with any centre and radius.

Q10. Prove that vertical angles are equal.

Solution:

• When two lines intersect, opposite angles are equal.

• This follows from Euclid’s axioms.

Q11. Prove that if two lines are perpendicular to the same line, they are parallel.

Solution:

• Both lines make right angles with the same line.

• Hence, they are parallel.

Q12. Prove that if equals are multiplied by equals, the products are equal.

Solution:

• Let $a=b$, $c=d$.

• Then $ac=bd$.

• Hence, products are equal.

Q13. Prove that if equals are divided by equals, the quotients are equal.

Solution:

• Let $a=b$, $c=d$.

• Then $\frac{a}{c}=\frac{b}{d}$.

• Hence, quotients are equal.

Q14. Prove that two distinct parallel lines never meet.

Solution:

• By Euclid’s fifth postulate, parallel lines do not intersect.

Q15. Prove that if two straight lines intersect, the sum of adjacent angles is 180°.

Solution:

• Straight angle = 180°.

• Adjacent angles on a straight line add up to 180°.

FAQs (10 for Exercise 5.2)

1. Q: What is Euclid’s first axiom? A: Things equal to the same thing are equal to one another.

2. Q: What is Euclid’s second axiom? A: If equals are added to equals, wholes are equal.

3. Q: What is Euclid’s third axiom? A: If equals are subtracted from equals, remainders are equal.

4. Q: What is Euclid’s fourth postulate? A: All right angles are equal to one another.

5. Q: What is Euclid’s fifth postulate? A: Known as the parallel postulate.

6. Q: Why are axioms important? A: They are universal truths used in proofs.

7. Q: Why are postulates important? A: They are assumptions specific to geometry.

8. Q: What is the difference between axioms and postulates? A: Axioms are general truths; postulates are geometry‑specific.

9. Q: What is the logical basis of geometry? A: Proofs derived from axioms and postulates.

10. Q: Why study Euclid’s geometry today? A: It builds logical reasoning and forms the foundation of mathematics.

Conclusion

Exercise 5.2 has 15 questions that apply Euclid’s axioms and postulates to basic proofs. This strengthens logical reasoning and prepares you for advanced geometry.

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