MATH 6118-090
Non-Euclidean Geometry

Exercise Set #2

1.      Suppose f is an isometry and suppose there exist two distinct points P and Q such that  and . Show that f is either the identity or a reflection.

2.      Prove that if a line  is sent to itself under a reflection through , then  and  intersect at right angles.

3.      Suppose that f and g are two isometries such that  and , and  for some nondegenerate triangle . Show that . That is, show that  for any point P.

4.      Prove the Star Trek lemma for an acute angle for which the center O is outside the angle.

5.      (Bow Tie Lemma) Let A, A', B and C lie on a circle, and suppose  and  subtend the same arc. Show that .

Text Box: Figure 1

Text Box: Figure 2

6.      In Figure 1, if , what is the angle at D?

7.      Suppose that two lines intersect at P inside a circle and meet the circle at A and A' and at B and B', as shown in Figure 2. Let  and  be the measures of the arcs  and  respectively. Prove that

.

8.      Suppose an angle  is defined by two rays which intersect a circle at four points. Suppose the angular measure of the outside arc it subtends is  and the angular measure of the inside arc it subtends is .(So in Figure 3,  and .)  Show

Figure 3