Prisms. Take any polygon \(A B C D E\) (Figure 39 ), and through its vertices,
draw parallel lines not lying in its plane. Then on one of the lines, take any
point \(\left(A^{\prime}\right)\) and draw through it the plane parallel to the
plane \(A B C D E\), and also draw a plane through each pair of adjacent
parallel lines. All these planes will cut out a polyhedron \(A B C D E
A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\) called a prism.
The parallel planes \(A B C D E\) and \(A^{\prime} B^{\prime} C^{\prime}
D^{\prime} E^{\prime}\) are intersected by the lateral planes along parallel
lines (\$13), and therefore the quadrilaterals \(A A^{\prime} B^{\prime} B, B
B^{\prime} C^{\prime} C\), etc. are parallelograms. On the other hand, in the
polygons \(A B C D E\) and \(A^{\prime} B^{\prime} C^{\prime} D^{\prime}
E^{\prime}\), corresponding sides are congruent (as opposite sides of
parallelograms), and corresponding angles are congruent (as angles with
respectively parallel and similarly directed sides). Therefore these polygons
are congruent.
Thus, a prism can be defined as a polyhedron two of whose faces are congruent
polygons with respectively parallel sides, and all other faces are
parallelograms connecting the parallel sides. The faces \(\left(A B C D
E\right.\) and \(\left.A^{\prime} B^{\prime} C^{\prime} D^{\prime}
E^{\prime}\right)\) lying in parallel planes are called bases of the prism. The
perpendicular \(O O^{\prime}\) dropped from any point of one base to the plane
of the other is called an altitude of the prism. The parallelograms \(A
A^{\prime} B^{\prime} C, B B^{\prime} C^{\prime} C\), etc. are called lateral
faces, and their sides \(A A^{\prime}, B B^{\prime}\), etc., connecting
corresponding vertices of the bases, are called lateral edges of the prism.
The segment \(A^{\prime} C\) shown in Figure 39 is one of the diagonals of the
prism.
A prism is called right if its lateral edges are perpendicular to the bases
(and oblique if they are not). Lateral faces of a right prism are rectangles,
and a lateral edge can be considered as the altitude.
A right prism is called regular if its bases are regular polygons. Lateral
faces of a regular prism are congruent rectangles.
Prisms can be triangular, quadrangular, etc. depending on what the bases are:
triangles, quadrilaterals, ete.
