91Ó°ÊÓ

Given the hypothesis of the acute angle, both Saccheri and Lambert showed that the sum of the angles of any triangle is less than two right angles. Let the difference between \(180^{\circ}\) and the angle sum of a triangle be the defect of the triangle. Suppose triangle \(A B C\) is split into two triangles by line \(B D\) (Fig. 20.16). Show that the defect of triangle \(A B C\) is equal to the sum of the defects of triangles \(A B D\) and \(B D C\).

Show that an "Euler path" over a series of bridges connecting certain regions (a path that crosses each bridge exactly once) is always possible if there are either two or no regions that are approached by an odd number of bridges.

Find the numbers of vertices, edges, and faces for each of the five regular polyhedra and confirm that Euler's formula holds in these five cases.