NON EUCLIDEAN GEOMETRY

A quadrilateral ABCD is convex if the two vertices on each edge are on the same side of the line through the opposite edge, e.g., A and B lie on the same side of the line CD, B and C lie on the same side of the line DA, etc.

A quadrilateral ABCD is a parallelogram if the lines through opposite edges do not intersect, that is, the lines AB and CD do not intersect, and the lines BC and AD do not intersct.

1. Prove the following:
2. Claim: Every parallelogram on E² or H² is convex. ( E is euclidean plane; H is hyperbolic plane)
3. Prove the following:
4. Claim: Let ABX be a triangle on E² or H². Let C be in the interior of BX, and let D be in the interior of AX. The quadrilateral ABCD is convex.