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 $ \mathrm{AB}=\mathrm{PQ} $ and $ \mathrm{PQ}=\mathrm{XY} $, then $ \mathrm{AB}=\mathrm{XY} $.
Fig. $ 5.9 $
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To do:

We have to find which of the given statements are true and which are false.

Solution:

(i) We know that,

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

For instance,

Therefore,

The statement, that only one line can pass through a single point is False.

(ii)  We know that,

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

For instance, $P$ and $R$ are two points.

Therefore,

The statement, that there are an infinite number of lines which pass through two distinct points is False.

(iii) We know that,

A line can be extended on both sides infinitely.

Since a line is terminated it can be indefinitely produced on both sides.

For instance,

(iv) We know that,

When the two circles are equal the circumference and the centres of both the circles coincide with each other.

Therefore,

The two circles may have equal radii.

Hence, the statement that if two circles are equal, then their radii are equal is True.

(v) We know that,

According to Euclid's First axiom: The things which are equal to the same thing are also equal to one another

Therefore, 

If $AB=PQ$ and $PQ=XY$, then $AB=XY$.

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Updated on: 10-Oct-2022

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