Part of the exam will be multiple choice, part will be fill in the blanks and part will be drawing diagrams.

You must be able to write the first line of a proof by contradiction. This will be tested by multiple choice.

You must be able to fill in the justifications of a proof which uses axioms and definitions. This will be tested by filling in the blanks.

You may be tested through drawing and/or multiple choice:

• Draw/identify lines between points on Hyperbolic space, euclidean space and the sphere.
• Find/identify parallel lines on euclidean space and the sphere.
• Figure out angles on euclidean space, hyperbolic space and the sphere (approximately) and find sums of the angles of triangles.
• Find congruent triangles in a given diagram using SAS SSS or ASA.
• Identify isosceles triangles in all three spaces (warning in hyperbolic can only check for matching angles).
• You must be able to provide pictures demonstrating on the sphere why certain axioms fail there. Similarly have an idea of why axioms fail on the rational plane, the space of three points, the space of one point and the real line.
• You must be able to find the sup and the inf of a set of real numbers.

You must be able to identify if a set is convex in Euclidean and hyperbolic space. You must be able to show that a set is not convex by drawing a line segment which demonstrates this fact (in euclidean and in hyperbolic space).

You must be able to find the mistake in an incorrect proof of a false statement. This will be tested by multiple choice.

This exam is going to focus on ideas and concepts not the rigorous difficult proofs that have been in the Proving Projects. Good luck!