Name: Euclid’s Fifth Postulate and the Equivalent
Uploaded: 2019-07-19
Duration: 8 min 53 s
Description: In this video, let's learn about Euclid’s Fifth Postulate and its significance. We will also learn about PLAYFAIR’S AXIOM, which is an equivalent form of the...

Question

Using postulate

5

show that there exists a line parallel to a given line.

Accepted Answer

Euclid's $5^{th}$ postulate states that 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.

This implies that if $n$ intersects lines $l$ and $m$ and if $∠ 1 + ∠ 2 < 180 °$ then $∠ 3 + ∠ 4 > 180 °$ . In that case, producing line $l$ and $m$ further will meet in the side of $∠ 1$ and $∠ 2$ which is less than $180 °$ .

If $∠ 1 + ∠ 2 = 180 °$ , then $∠ 3 + ∠ 4 = 180 °$ .

In this case, the lines $l$ and $m$ neither meet at the side of $∠ 1$ and $∠ 2$ nor at the side of $∠ 3$ and $∠ 4$ .

Implying that the lines $l$ and $m$ will never intersect each other.

Therefore, the lines are parallel.

Using postulate $5$ show that there exists a line parallel to a given line.

Important Questions on Introduction to Euclid's Geometry

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