DO THE CONSTRUCTIONS FOR EACH. NOTE: THAT A COUNTEREXAMPLE EXISTS FOR EACH.

Assuming neutral geometry, the following statements are equivalent to the Parallel Axiom. For each statement, construct a counterexample in the Poincar disk and explain how it is a counterexample. Note: Everywhere it says "straight line", you should interpret it as "H-line" since we're working in the Poincar disk now. 1. Parallel Axiom (Playfair's Axiom). Through a point not on a given straight line, there exists at most one straight line parallel to the given line. 2. Parallel Postulate (Euclid's 5th Postulate). If a straight line falling on two straight lines makes the interior angles on the same side add up to less than two right angles, then the two straight lines meet on the side on which the angles are less than two right angles. 3. Converse of Alternate Interior Angles Theorem (Euclid I.29). A straight line falling on two parallel lines makes the alternate interior angles equal to each other. 4. Transitive Property of Parallels (Euclid I.30). Straight lines parallel to the same straight line are parallel to each other. 5. Proclus' Axiom. If a line cuts one of two parallel lines, then it also cuts the other. Use color and line style to enhance the clarity of your diagrams. Include captions of how each diagram is a counterexample