Lines l and m are parallel. t is a transversal of l and m. t is perpendicular to l. Playfair's postulate states the for every line l and point p then there is a unique line parallel to l through p. Need to show that t is perpendicular to m.[br][br]Let p be the poinnt where t intersects m. By the converse of the alternate interior angle theorem we have parallel lines cut by a transversal thus the alternate interior angles must be congruent. Since the angles between t and l are 90 degrees by construction then the angles between t and m must be 90 degrees and t is perpendicular to m.[br][br]Conversely, given t a transversal of l and m. where l and m are parallel. t is parallel to l and t is parallel to m.[br]Need to show for any line l and for all points P there is a unique line parallel to l through P. Let P be a point and drop a perpendicular from P to l, call it t. Let m be a line through P perpendicular to t by the converse of the statement we have that m and l are parallel. How do we know that m is unique? By the converse of the Alternate Interior Angle theorem. If there was another line parallel to l through P it would have to have the same alternate interior angles so this "other line" would be perpendicular to t which means it's m.