# 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 following figures,if AB = PQ and PQ = XY, then AB= XY.

(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 following figures,if AB = PQ and PQ = XY, then AB= XY.

**Solution:**

(i) False. Infinite number of lines pass through a single point .

(ii) False. There is unique line that passes through two distinct points

(iii) True. Postulate 2.

(iv) True. If we superimpose the region bounded by one circle on the other, then they coincide. So, their centres and boundaries coincide therefore, their radii will coincide.

(v) True. The first Axiom of Euclid.

# 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.

(i) Parallel lines (ii) Perpendicular Lines (iii) line segment

(iv) radius of a circle (v) square.

**Solution:**

(i) Parallel lines. Lines which do not intersect each other anywhere are called parallel lines.

(ii) Perpendicular lines.

**Two lines which are at a right angle to each other are called perpendicular lines.**

(iii) Line segment.

**It is a terminated line.**

(iv) Radius.

**The length of the line-segment joining the centre of a circle to any point on**

its circumference is called its radius.

(v) Square. All the sides are equal in length.

# Question-3

**Consider two â€˜Postulatesâ€™ given below:**

a. Given any two distinct points A and B, there exists a third point C which is in between A and B.

b. 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 Euclidâ€™s Postulates? Explain.

a. Given any two distinct points A and B, there exists a third point C which is in between A and B.

b. 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 Euclidâ€™s Postulates? Explain.

**Solution:**

a. Yes! These postulates contain two undefined terms: Point and Line.

b. Yes ! These postulates are consistent because they deal with two different situations (i) say that given two points A and B, there is a point C lying on the line in between them, (ii) say that given two points A and B, we can take another point C not lying on the line through A and B. These 'postulates' do not follow from Euclid's postulate however, they follow from Axiom 5.1.

# Question-4

**If a point C lies between two points A and B such that AC = BC, then prove that AC =AB. Explain by drawing the figure.**

**Solution:**

AC = BC

AC + AC = BC + AC |Equals are added to equally

â‡’ 2AC = AB | BC + AC coincides with AB

â‡’ AC =

# Question-5

**In previous question, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.**

**Solution:**

In the previous question we proved that,

â‡’ AC = -------(1) (

**âˆµ**C be a point which is lies between A and B)

Suppose D be mid point of A and B,

â‡’ AD = --------(2)

From (1) and (2),

AC = AD

âˆ´ Every line segment has one and only one mid-point.

# Question-6

**Solution:**

AC = BD |Given â€¦â€¦â€¦.(1)

AC = AB + BC | Point B lies between A and C â€¦..(2)

BD = BC + CD | Point C lies between B and D â€¦..(3)

Substituting (2) and (3) in (1), we get

AB + BC = BC + CD â‡’ AB = CD |Subtracting equals from equals

# Question-7

**Why is axiom5, in the list of Euclidâ€™s axioms, considered a â€˜universal truthâ€™ ? (Note that the question is not about the fifth postulate.)**

**Solution:**

This is true for any thing in any part of the world, this is a universal truth.

# Question-8

**How would you rewrite Euclidâ€™s fifth postulate so that it would be easier to understand?**

**Solution:**

Two distinct intersecting lines cannot be parallel to the same line.

# Question-9

**Does Euclidâ€™s Fifth postulate imply the existence of parallel lines? Explain.**

**Solution:**

If a straight line

*l*falls on two straight lines

*m*and

*n*such that sum of the interior angles on one side of

*l*is two right angles, then by Euclidâ€™s fifth postulate the line will not meet on this side of

*l*. Next, we know that the sum of the interior angles on the other side of line

*l*will also be two right angles.

Therefore, they will not meet on the other side also. So, the lines

*m*and

*n*never meet and are, therefore parallel.