Problem 4. (20 pts) Triangles in S2 (Part I). Let TCS be a triangle in S, defined by the lines L1, L2, L3 CS. (a) Show that the area of the unit radius 2-sphere is 4. (b) Show that the area of a sector of angle a is a/2 the area of the 2-sphere. A sector of angle a is the bigon described by two lines at angle a. (c) Show that the complement S2 {L1, L2, L3} consists of eight triangular regions, where L1, L2, L3 CS2 are lines defining a triangle. (d) Show that there exist six pairs of such regions such that the union of the pair of region is a sector as in Part (b). (e) Let T S2 be a triangle, and a, 8, 7 be the interior angles of T. Show that a + 8 + = + Area(T).

Problem 4. (20 pts) Triangles in S2 (Part I). Let TCS be a triangle in S, defined by the lines L1, L2, L3 CS. (a) Show that the area of the unit radius 2-sphere is 4. (b) Show that the area of a sector of angle a is a/2 the area of the 2-sphere. A sector of angle a is the bigon described by two lines at angle a. (c) Show that the complement S2 {L1, L2, L3} consists of eight triangular regions, where L1, L2, L3 CS2 are lines defining a triangle. (d) Show that there exist six pairs of such regions such that the union of the pair of region is a sector as in Part (b). (e) Let T S2 be a triangle, and a, 8, 7 be the interior angles of T. Show that a + 8 + = + Area(T).