State whether the following is true or false.

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

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True

False

answer is A.

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Detailed Solution

Given Euclid’s fifth postulate.
We are asked that whether Euclid's fifth postulate implies the existence of parallel lines or not.
According to Euclid's fifth postulate, ''if a straight line falling on two straight lines, making interior angles on some side which are less than two right angles, then the two straight lines, if produced indefinitely, then they meet on that side on which the angles are less than two right angles.''
Consider the two parallel lines q and r and a line p which passes through both of the lines as shown in the figure given below.
Question Image

Now we use prove the existence of parallel lines using Euclid's fifth postulate. In the figure, the lines q and r are parallel to each other.
Then we must have,

∠ 1 + ∠ 2 = 180 °

Since the angle is not less than

180 °

, by the Euclid's fifth postulate, then the lines q and r will not meet on left side of the line p.
Similarly, we must have,

∠ 3 + ∠ 4 = 180 °

Since the angle is not less than

180 °

, by the Euclid's fifth postulate, then the lines q and r will not meet on right side of the line p.
Hence, the Euclid's fifth postulate implies the existence of parallel lines.
Therefore, option (1) is correct and the statement is true.

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