## Chapter 3 Introduction to Euclid’s Geometry Ex 3.1

**Question 1.** **Which of the following statements are true and which are false? Give reasons for your answers.** **(i) Only one line can pass through a single point.** **(ii) There are an infinite number of lines which pass through two distinct points.** **(iii) A terminated line can be produced indefinitely on both the sides.** **(iv) If two circles are equal, then their radii are equal.** **(v) In figure, if AB – PQ and PQ = XY, then AB = XY.**

**Solution:** **(i)** False. In a single point, infinite number of lines can pass through it.

**(ii)** False. For two distinct points only one straight line is passing.

**(iii)** True.

**(iv)** True. [∵ Radii of congruent (equal) circles are always equal] **(v)** True. AB = PQ …..(i)

PQ = XY

⇒ XY = PQ …(ii)

From Eqs. (i) and (ii), we get AB = XY

**Question 2.** **Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they and how might you define them?** **(i) Parallel lines** **(ii) Perpendicular lines** **(iii) Line segment** **(iv) Radius of a circle** **(v) Square** **Solution:** **(i)** Parallel lines Two lines in a plane are said to be parallel, if they have no point in common.

In figure, x and y are said to be parallel because they have no point in . common and we write, x∥

y.

Here, the term point is undefined. **(ii)** Perpendicular lines Two lines in a plane are said to be perpendicular, if they intersect each other at one right angle.

In figure, P and Q are said to be perpendicular lines because they ; intersect each other at 90° and we write Q ⊥ P.

Here, the term one right angle is undefined. **(iii)** Line segment The definite length between two points is called the line segment.

In figure, the definite length between A and B is line represented by AB.

Here, the term definite length is undefined. **(iv)** Radius of a circle The distance from the centre to a point on the circle is called the radius of the circle.

In the adjoining figure OA is the radius.

Here, the term, point and centre is undefined. **(v)** Square A square is a rectangle having same length and breadth.

Here, the terms length, breadth and rectangle are undefined.

**Question 3.**
**Consider two ‘postulates’ given below**
**(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.**
**(ii) There exist atleast three points that are not on the same line.**
**Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain**.
**Solution:**

There are so many undefined words which should be knowledge. They are consistent because they deal with two different situations that is

(i) if two points A and B are given, then there exists a third point C which is in between A and B.

(ii) if two points A and B are given, then we can take a point C which don’t lie on the line passes through the point A and B.

These postulates don’t follow Euclid’s postulates. However, they follow axiom Euclid’s postulate 1 stated as through two distinct points, there is a unique line that passes through them.

**Question 6.** **In figure, if AC = BD, then prove that AB = CD.**

**Solution:**

We have

AC = BC

⇒ AC – BC = BD – BC (∵ Equals are subtracted from equals)

⇒ AB = CD

**Question 7.** **Why is axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’?** **(Note that, the question is not about the fifth postulate.)** **Solution:**

According to axiom 5, we have The whole is greater than a part, which is a universal truth.

Let a line segment PQ = 8 cm. Consider a point R in its interior, such that PR = 5 cm

Clearly, PR is a part of the line segment PQ and ft lies in its interior.

⇒ PR is smaller than PQ.

Hence, the whole is greater than its part.