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Prove that the following statement is equivalent to euclid's 5th postulate, 

If a straight line intersects one of two parallel lines, it will always intersect the other.

Suppose that the lines l and k are parallel and line m intersects k at a point. We will use the playfair form of the Euclidean fifth postulate.
If m does not intersect l, then m is parallel to l. This would mean that there are two parallels to l through P(m and k). contradicting play fair's axiom. Thus, m must also intersect l. Now let P be a point and l a line not through P. we can construct a line parallel to l through P, call it k. Let m be any other line through P. Since it intersects one of two parallel lines (k), it must intersect the other (l) by assumption. thus, there is only one line through P which is parallel to l.



Which  of the following statemants are true and which are false?
(i) A straight line can be drawn from one point to another point.
(ii) The edges of the surfaces are points.
(iii) Infinitely many lines can be drawn through two points.
(iv) Two lines can have only one point in common.

(i) True.
(ii) False, the edges of the surfaces are lines.
(iii)False, only one line can be drawn through two given points.

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