Will two parallel lines with a transversal have complementary angles

No since a complimentary angle is when the two angles add together to make 90°. a transversal is like a line cutting through the two parallel lines so it is not possible for it to have complimentary angles.

PS. Is this confusing?

Two angles are said to be supplementary when the sum of the two angles is 180 degrees. When a transversal intersects with two parallel lines eight angles are produced. The eight angles will together form four pairs of corresponding angels.