MATH 402 Worksheet 2

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MATH 402 Worksheet 2
Friday 9/2/16
(1) Prove the vertical angle theorem within Hilbert’s axiomatic system. Namely, let l, m be lines
intersecting at P . Show that the vertical angles formed by l and m are equal to each other.
(2) The following theorem is called the exterior angle theorem: Given a triangle 4ABC, extend one
of its sides, for example AC to D. Then, the exterior angle produced (i.e. ∠DAB) is greater than
either of the two opposite interior angles (i.e. ∠ABC and ∠ACB).
Euclid gave the following proof:
Why is this proof flawed within the Euclidean system? What can be done to salvage it in
Hilbert’s system?
(3) Prove the following converse to the parallel postulate (without using the parallel postulate): Let
l, m be two lines, and n a third line intersecting both l, m, such that the alternate interior angles
formed are congruent. Then l and m are parallel.
(4) Formally define congruence of triangles.
(5) Euclid proved the side-angle-side (SAS) congruence, but there are issues with his proof. In contrast, Hilbert assumed SAS as an axiom. Using Hilbert’s axioms, state and prove the other three
congruence rules: ASA, AAS, SSS.

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