NCERT Solutions Ch- 5 Introduction to Euclid Geometry

Ex 5.1

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 Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.


(i) False

There can be infinite number of lines that can be drawn through a single point.

(ii) False

Through two distinct points there can be only one line that can be drawn.

(iii) True

A line that is terminated can be indefinitely produced on both sides as a line can be extended on both its sides infinitely.

(iv) True

The radii of two circles are equal when the two circles are equal. The circumference and the centre of both the circles coincide; and thus, the radius of the two circles should be equal.

(v) True

According to Euclid’s 1st axiom- “Things which are equal to the same thing are also equal to one another”.


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


Yes, there are other terms which need to be defined first, they are:

Plane: Flat surfaces in which geometric figures can be drawn are known are plane. A plane surface is a surface which lies evenly with the straight lines on itself.

Point: A dimensionless dot which is drawn on a plane surface is known as point. A point is that which has no part.

Line: A collection of points that has only length and no breadth is known as a line. And it can be extended on both directions. A line is breadth-less length.

(i) Parallel lines – Parallel lines are those lines which never intersect each other and are always at a constant distance perpendicular to each other. Parallel lines can be two or more lines.

(ii) Perpendicular lines – Perpendicular lines are those lines which intersect each other in a plane at right angles then the lines are said to be perpendicular to each other.

(iii) Line Segment – When a line cannot be extended any further because of its two end points then the line is known as a line segment. A line segment has 2 end points.

(iv) Radius of circle – A radius of a circle is the line from any point on the circumference of the circle to the center of the circle.

(v) Square – A quadrilateral in which all the four sides are said to be equal and each of its internal angle is right angles is called square.

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.
Yes, these postulates contain undefined terms such as ‘Point and Line’. Also, these postulates are consistent because they deal with two different situations as
(i) says that given two points A and B, there is a point C lying on the line in between them. Whereas
(ii) says that, given points A and B, you can take point C not lying on the line through A and B.
No, these postulates do not follow from Euclid’s postulates, however they follow from the axiom, “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 = frac { 1 }{ 2 } AB, explain by drawing the figure.

AC = BC [Given]
∴ AC + AC = BC + AC
[If equals added to equals then wholes are equal]
or 2AC = AB [∵ AC + BC = AB]
or AC = frac { 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.
Let the given line AB is having two mid points ‘C’ and ‘D’.

AC = frac { 1 }{ 2 } AB ……(i)
and AD = frac { 1 }{ 2 } AB ……(ii)
Subtracting (i) from (ii), we have
AD – AC = frac { 1 }{ 2 } AB-frac { 1 }{ 2 } AB
or AD – AC = 0 or CD = 0
∴ C and D coincide.
Thus, every line segment has one and only one mid-point.

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

Given: AC = BD
⇒ AB + BC = BC + CD
Subtracting BC from both sides, we get
AB + BC – BC = BC + CD – BC
[When equals are subtracted from equals, remainders are equal]
⇒ AB = CD


Ex 5.2

1. How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?


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.

i.e., the Euclid’s fifth postulate is about parallel lines.

Parallel lines are the lines which do not intersect each other ever and are always at a constant perpendicular distance apart from each other. Parallel lines can be two or more lines.

A: If X does not lie on the line A then we can draw a line through X which will be parallel to that of the line A.

B: There can be only one line that can be drawn through the point X which is parallel to the line A.

2. Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.


Yes, Euclid’s fifth postulate does imply the existence of the parallel lines.

If the sum of the interior angles is equal to the sum of the right angles, then the two lines will not meet each other at any given point, hence making them parallel to each other.

∠1+∠3 = 180o

Or  ∠3+∠4 = 180o

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