Mathematics Equivalent Versions of Euclid’s Fifth Postulate

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Equivalent Versions of Euclid’s Fifth Postulate

Equivalent Versions of Euclid’s Fifth Postulate

We see that by implication, no intersection of lines will take place when the sum of the measures of the interior angles on the same side of the falling line is exactly 180°.

There are several equivalent versions of this postulate. One of them is ‘Playfair’s Axiom’, as stated below

‘For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l’.

From Fig. 5.11, you can see that of all the lines passing through the point P, only line m is parallel to line l.

This result can also be stated in the following form:

bb"Two distinct intersecting lines cannot be parallel to the same line."
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Consider the following statement : There exists a pair of straight lines that are everywhere equidistant from one another. Is this statement a direct consequence of Euclid’s fifth postulate? Explain.
Class 9 Chapter 5 Example 3
Solution:

Take any line l and a point P not on l. Then, by Playfair’s axiom, which is equivalent to the fifth postulate, we know that there is a unique line m through P which is parallel to l.

Now, the distance of a point from a line is the length of the perpendicular from the point to the line. This distance will be the same for any point on m from l and any point on l from m. So, these two lines are everywhere equidistant from one another.

Remark :

The geometry that you will be studying in the next few chapters is Euclidean Geometry. However, the axioms and theorems used by us may be different from those of Euclid’s