I need help with question 8.

LEGENDRE GETS IT RIGHT = MA391_HW3.pdf 1 / 1 110% + [ 210 MA 391 ASSIGNMENT #3 Figure 10.6. A lune on a sphere (left) and the view from above (right). H A E a D Answers to this assignment may be handwritten or typed. Consider Legendre's proof of Euler's formula for polyhedra. The explanation given in Euler's Gem leaves many questions unanswered, which we will explore in this assignment. One of the main goals of this assignment is to demonstrate how easy it is to overlook "gaps" in well-written mathematical exposition. All of the following questions are natural and important, but it is easy to read Euler's Gem without a critical eye and hence miss these gaps. Problem 8: Provide a detailed proof of the claim on page 93 of Euler's Gem that triangles ADE and AGH can be combined to form a lune of angle a. Problem 9: On page 93, it is claimed that on a spherical polygon, the area of the polygon can be obtained by labeling each angle with its radian measure, each edge with TT, and the face with 27, then summing these values. Verify that this claim is valid. Figure 10.7. Great circles in a hemisphere. Now, consider a geodesic triangle ABC on a unit sphere with interior angles a, b, and c. The triangle is contained in some hemisphere. Extend the sides of ABC to meet the boundary of this hemisphere. As in figure 10.7. let D, E, F, G, H, and I denote the points where these lines meet the hemisphere's edge By the symmetries of the sphere, the sum of the areas of the regions ADE and AGH is the same as the area of a lune with angle a. In other