  We use a probabilistic interpretation of solid angles to generalize the
well-known fact that the inner angles of a triangle sum to 180 degrees. For the
3-dimensional case, we show that the sum of the solid inner vertex angles of a
tetrahedron T, divided by 2*pi, gives the probability that an orthogonal
projection of T onto a random 2-plane is a triangle. More generally, it is
shown that the sum of the (solid) inner vertex angles of an n-simplex S,
normalized by the area of the unit (n-1)-hemisphere, gives the probability that
an orthogonal projection of S onto a random hyperplane is an (n-1)-simplex.
Applications to more general polytopes are treated briefly, as is the related
Perles-Shephard proof of the classical Gram-Euler relations.
