# Chapter 44: Chapter 44

Problem 812

Show that the lateral edges of a regular pyramid are congruent.

Problem 813

If two pyramids have congruent altitudes and bases with equal areas, show that sections parallel to the bases at equal distances from the vertices have equal area.

Problem 814

Show that the perpendicular from the vertex to the base of a regular pyramid contains only points that are equidistant from the faces.

Problem 815

Given that the perpendicular from the vertex to the base of a regular pyramid contains only points equidistant from the faces, show the converse. That is, show that if a point in the interior of a regular pyramid is equidistant from the faces, then it lies on the perpendicular drawn from the vertex.

Problem 818

Show that the angle sum of a spherical triangle is greater than \(180^{\circ}\) and less than \(540^{\circ}\).

Problem 819

On a sphere of radius 9 inches, the perimeter of a spherical triangle is $12 \pi\( inches. The sides of the triangle are in the ratio \)3: 4: 5$ a) Find the sides of the triangle in degrees. b) Find the angles of its polar triangle. c) Find the area of the polar triangle in square inches, (answer may be left in terms of \(\pi\).) d) A zone on this sphere is equal in area to the polar triangle. Find the number of inches in the altitude of the zone.

Problem 820

The number of degrees in the angles of a spherical triangle are in the ratio of \(3: 4: 5\). The area of the triangle is equal to the area of a zone of altitude 3 on the same sphere. If the radius of the sphere is 15 , find each angle of the triangle.

Problem 821

Show that if a point on a sphere is at a distance of a quad rant from each of two other points on the sphere, not the extremities of a diameter, then the point is a pole of the great circle passing through these points.

Problem 822

Given two polar triangles, show that each angle of one polar triangle has the same measure as the supplement of the side lying opposite it in the other.

Problem 823

Show that if one spherical triangle is the polar triangle of another, then the second is the polar triangle of the first.