9 Questions on Non Euclidean Geometry - Exam | MATH 402, Exams of Geometry

Download 9 Questions on Non Euclidean Geometry - Exam | MATH 402 and more Exams Geometry in PDF only on Docsity! Name: Math 402 - Final Exam - May 8, 2007 Time: 3 hours. Write your answers on the exam. You may not use any books or notes other than the page of theorems provided. There are 100 points possible. 1. (5 points) Give the definition of isometry. 2. (10 points) Prove that if f and g are isometries, then their composition f ◦ g is also an isometry. 3. (10 pages) List all the types of isometries. 6. (5 points each part) How, if at all, does hyperbolic geometry differ from Euclidean geometry in each of the following categories? (a) parallel lines (b) Side-Angle-Side Congruence Theorem for triangles (c) similar triangles 7. (10 points) Is it possible or not to tile (tessellate) each of the following spaces with equilateral triangles with eight triangles meeting at each vertex? Explain briefly. (a) Euclidean space (b) hyperbolic space (c) the sphere 8. (10 points) Recall that in hyperbolic space, there is a constant k such that for any triangle, area(4ABC) = k2defect(4ABC). Use this equation to find numbers m and M such that m ≤ area(4ABC) ≤M for all triangles in hyperbolic space. Is there any 4ABC which actually has area m or M? Explain.