MATH 213 Final, Spring 1997

haga LO uu yur Pea NWOREAINS AMERICAN UNIVERSITY OF BEIRUT ==" MATHEMATICS 213, FINAL EXAMINATION SECOND SEMESTER, 1996-97 Answer the following questions: I. (a) State the ruler postulate. (3 pts.) (b) Use the ruler postulate to prove that (i) of any three collinear points exactly one lics between the oller two. (6 pts.) (ii) if four collinear points A, B,C and D satisfy A- B—C and B-—C-D, then they satisfy A-— B- DandA-C-D. (6 pts.) II. Use the plane-separation postulate and the postulate of Pasch to prove that for any triangle AABC (a) the interiors of the segments BC, CA and AB have no transversal, and that (5 pts.) (b) the rays BC \BC, AC \AC and AB \AB always have a transversal. (5 pts.) Ill. (a) Define a convex quadrilateral. (3 pts.) (b) Use the Crossbar Theorem to Show that the diagonals of a convex quadrilateral intersect each other. (G pts.) (c) Show that no full line can be contained in the interior of a triangle. (6 pts.) IV. (a) State the SAS Postulate. (3 pts.) (b) Prove the AS'A Theorem from the SAS Postulate. (6 pts.) (c) Use the exterior angle Theorem to prove that for a given triangle |. % AABC with a point D satisfying B- D-C, either AD < AB or AD < AC. (6 pts.) V. (a) Define a parallel projection. (3 pts.) (b) Show that parallel projections preserve betweenness. (6 pts.) (c) Give Euclid’s proof of the Basic Similarity Theorem which uses the equivalence of areas of two triangles having equal bases and altitudes. (6 pts.) VI. (a) Show that isometry is an equivalence relation. (3 pts.) (b) Prove that if a set M is isometric to the unit interval [0,1], then M is a unit interval. (6 pts.) (c) Show that a dilation preserve circles. (6 pts.) VII. (a) Use an unmarked ruler and a collapsible compass to divide a segment into n congruent segments. (6 pts.) (b) Define the Surd field. (3 pts.) (c) Is 3/9 a Surd number? Justify your answer. (3 pts.)