Lesson: Introduction to Euclid's Geometry

Exercise 5.1 (7)

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

By Euclid’s first postulate, a straight line may be

drawn from anyone point to any other point.

Suppose, there are points; A, B, C, and D in a plane.

Now, we can draw lines from point A to B, A to C,

and A to D. It proves that many lines can pass

through a single point.

(ii) False.

Let us assume that there are two points P and Q in a

plane. Now, join points P and Q as shown.

It is clear from the figure that only one line can pass

From the above figure, it is clear that a terminated line

can be produced indefinitely.

(iv) True.

Two circles will coincide if we super impose the

region bounded by one circle on the other. Their

centers and boundaries will also coincide. Therefore,

their radii will be equal.

(v) True. By Euclid’s first axiom, things which are

equal to the same thing are equal to one another.

Hence, if AB = PQ and PQ = XY, then AB = XY.

Question: 2

Give a definition for each of the following terms. Are

there other terms that need to be defined first? What

Solution:

Yes, following are the terms that need to be defined

first:

Plane: A plane is a flat surface on which geometric

figures are drawn.

Point: A point is a dot drawn on a plane surface and is

dimensionless.

Line: A line is collection of points which can extend in

both directions and has only length not breadth.

Definitions:

(i) Parallel lines: If the perpendicular distance between

two lines is always constant, then they are parallel

lines.

(ii) Perpendicular lines: If two lines intersect each

other at right angle i.e. in a plane, then they are

(v) Square: A square is a quadrilateral whose all the

four sides are equal and each internal angle is a right

angle.

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 at least three points that are not on the

same line.

Do these postulates contain any undefined terms? Are

 Whether the points are in the same plane or not.

Yes, the two postulates are consistent as per the

following two situations.

Postulate i): Given any two distinct points A and B,

there exists a third point C which is in between A and

B.

This is consistent when there are two points A and B,

and a point C lying in between on the line segment

joining them.

Postulate (ii): (ii) There exist at least three points that

are not on the same line.

This is consistent when point C does not lie on the line

segment joining A and B.

Whether the postulate follow Euclid’s postulate:

The postulates do not follow from Euclid’s postulates.

Solution:

Here, AC = BC

Now, adding AC both the sides.

AC + AC = BC + AC (If equals are added to equals,

the wholes are equal.)

Also, BC + AC = AB (as it coincides with the line

segment AB)

∴ 2 AC = AB

Solution:

Let XY be the line segment and points A and B be two

different mid points of XY.

Now,

∴ A and B are midpoints of XY.

Therefore, XA = AY and XB = BY.

also, XA + AY = XY (as it coincides with the line

segment XY

Similarly, XB + BY = XY.

Now,

Similarly,

2 XB = XY --- (ii)

From (i) and (ii)

2 XA = 2 XB (Things which are equal to the same

thing are equal to one another.)

(Things which are double of the same

things are equal to one another.)

Thus, A and B are the same points. This contradicts

the fact that A and B are two different mid points of

XY. Thus, it is proved that every line segment has one

and only one mid-point.

Question: 6

In Fig. 5.10, if AC = BD, then prove that AB = CD.

Solution:

Given, AC = BD

From the figure,

AC = AB + BC

BD = BC + CD

According to Euclid's axiom, when equals are

subtracted from equals, remainders are also equal.

Subtracting BC from both sides,

Solution:

According to Axiom 5, the whole is always greater

than the part.

Let’s take an example of a bottle of milk. Suppose a

bottle of milk measures 1 liter. If a small amount of

milk is taken out, then the amount of milk left would

be less than the original amount of milk i.e.1 liter. So,

it is considered a ‘universal truth’.

Exercise 5.2 (2)