Q#1 Prove: Hilbert's Euclidean Parallel Postulate if and only if the opposite sides of a parallelogram are congruent. (A parallelogram is defined as a quadrilateral with pairs of opposite sides that are parallel.) note : this is if and only if question. so you have to assume that Hilbert's Euclidean Parallel Postulate holds and prove the the opposite sides of parallelogram are congruent valid. Then assume the opposite sides of parallelogram are congruent then prove Hilbert's Euclidean Parallel Postulate valid. Q#2 Prove: Hilbert's Euclidean Parallel Postulate if and only if the diagonals of a rhombus are perpendicular. (A rhombus is defined as a quadrilateral with four sides that are the same length.) note : this is if and only if question. so you have to assume that Hilbert's Euclidean Parallel Postulate holds and prove the diagonals of a rhombus are perpendicular valid. Then assume that the diagonals of a rhombus are perpendicular holds then prove Hilbert's Euclidean Parallel Postulate valid.