Math 300
Hour Exam #2
Name ________________
Tuesday, June 19
100 pts.
1. (15 pts.) Given quadrilateral ABCD with the line through AD perpendicular to the line through AB (at the point A), and the line through BC perpendicular to the line through AB (at the point B, and, finally, the line through DC perpendicular to the line through AD (at the point D), prove that the ray AC lies between the ray AD and the ray AB.
2. (15 pts.) Given triangle ABC with angles A and B acute. Use the exterior angle theorem to prove that the perpendicular from C to the line through AB lies on the segment AB.
3. (15 pts. - an essay question) Give an account of the continuity axioms. What do they accomplish? Draw on one of the homework problems to demonstrate that continuity is an issue.
4. Several questions on parallel lines (10 pts. each)
a. Hilbert's axiom states that, given a line and a point not on the line, no more than one line parallel to the given line passes through the given point. Why did Hilbert leave out the existence of at least one parallel line?
b. Use one of the equivalents to the parallel postulate to justify the statement "If line l is parallel to line m, and line m is parallel to line n, then either l = n or l is parallel to n.
c. What is a Saccheri quadrilateral? What was Saccheri trying to prove about quadrilaterals named after him? Why won't it work?
(problem 4 continued)
d. What was Clairaut's axiom? What would Clairaut's axiom imply about the angle sum of a triangle (and why)?
5. (15 pts.) Recall that a convex quadrilateral is one that has a pair of opposite sides such that each is contained in a half-plane bounded by the other. It can be shown (see problem 23 on page 141) that this implies that the other pair of opposite sides also has this property (each is contained in a half-plane bounded by the other). Assuming this fact, demonstrate that the diagonals of a convex quadrilateral intersect (Hint: use the crossbar theorem. This is problem 24 on page 141).