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
.
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