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

both the sides.

(iv) If two circles are equal, then their radii are equal.

(v) In Fig. 5.9, if AB = PQ and PQ = XY, then

AB = XY.

Solution:

(i) False.

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

through two distinct points.

(iii) True.

By Euclid’s postulate, a terminated line can be

produced indefinitely on both the sides.

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

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:

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

90°

said to be perpendicular to each other.

(iii) Line segment: A line segment is a part of a line

having two end points.

(iv) Radius of circle: It is the fixed distance between

the centre and the circumference of the circle.

(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

these postulates consistent? Do they follow from

Euclid’s postulates? Explain.

Solution:

Undefined terms in the postulates are:

Whether the point C lies on the line segment joining

AB or not.

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.

They follow the axiom that states that given two

distinct points, there is a unique line that passes

through them.

Question: 4

If a point C lies between two points A and B such that

AC = BC, then prove that AC = 1/2 AB. Explain by

drawing the figure.

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

AC 1/ 2 AB

Question: 5

In Question 4, point C is called a mid-point of line

segment AB. Prove that every line segment has one

and only one mid-point.

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,

XA + XA = AY + XA (If equals are added to equals,

the wholes are equal.)

--- (i)

2 XA XY

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

XA XB

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

AB BC BC CD

According to Euclid's axiom, when equals are

subtracted from equals, remainders are also equal.

Subtracting BC from both sides,

AB + BC BC = BC + CD − BC

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, 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)

Question: 1

How would you rewrite Euclid’s fifth postulate so that

it would be easier to understand?

Solution:

Euclid’s fifth Postulate:

If a straight line, falling on two straight lines, makes

the interior angles on the same side of it taken together

less than two right angles, then the two straight lines,

if produced indefinitely, meet on that side on which

the sum of angles is less than two right angles.

Interpreting the fifth postulate:

a) If two lines are parallel to each other and a line is

drawn intersecting the two parallel lines, then the

sum of the two interior angles that this line makes with

the parallel lines is equal to two right angles

.

(180 )

b) If a line intersects two lines, and the sum of the two

interior angles that this line makes with the parallel

lines is less than two right angles, then the two lines

that are intersected are not parallel to each other.

Question: 2

Does Euclid’s fifth postulate imply the existence of

parallel lines? Explain.

Solution:

Yes, Euclid’s fifth postulate implies the existence of

parallel lines.

If the sum of the interior angles is equal to the sum of

the two right angles then the two lines will not meet

each other on either sides and therefore, they will be

parallel to each other.

m and n will be parallel if,

1 3 180

Or

2 4 180