Of Planes.

THEOREM XI.

If two angles, not situated in the same plane, have their sides parallel and lying in the same direction, the angles will ba equal.

Again, since CE is equal and parallel to DF, CF will be a parallelogram, and EF will be equal and parallel to CD. Then, since AB and EF are both parallel to CD, they will be parallel to each other (Th. x); and since they are each equal to CD, they will be equal to each other. Hence, the figure BAEF is a parallelogram (Bk. I. Th. xxv), and consequently, AE is equal to BF. Hence, the two triangles ACE and BDF have the three sides of the one equal to the three sides of the other, each to each, and therefore the angle ACE is equal to the angle BDF (Bk. I. Th. viii).

THEOREM XII.

If two planes are parallel, a straight line which is perpendicula to the one will also be perpendicular to the other.

AB, BC, suppose the plane
ABC to be drawn, intersecting

the plane MN in the line AD: then, the intersection AD will be parallel to BC (Th. ix). But since AB is perpendicular to the plane NM, it will be perpendicular to the straight line AD, and consequently, to its parallel BC (Bk. I. Th. xii. Cor.)

In like manner, AB might be proved perpendicular to any other line of the plane PQ, which should pass through B; hence, it is perpendicular to the plane (Def. 1).

Cor. It from any point as H, any oblique lines, as HEF, HDC, be drawn, the parallel planes will cut these lines proportionally.

For, draw HAB perpendicular to the plane MN: then, by the theorem, it will also be perpendicular to PQ. Then draw AD, AE, BC, BF. Now, since AE, BF,

are the intersections of the plane

M

H

E

N

P

B

F

FHB, with the two parallel planes MN, PQ, they are paral

lel (Th ix.); and so also are AD, BC.

HA : HB : HE: HF,

Then,

GEOMETRY.

BOOK VI.

OF SOLIDS.

DEFINITIONS

1. Every solid bounded by planes is called a polyedron.

2. The planes which bound a polyedron are called faces. The straight lines in which the faces intersect each other, are called the edges of the polyedron, and the points at which the edges intersect, are called the vertices of the angles, or vertices of the polyedron.

3. Two polyedrons are similar, when they are contained by the same number of similar planes, and have their polyedral angles equal, each to each.

4. A prism is a solid, whose ends are equal polygons, and whose side faces are parallelograms.

Thus, the prism whose lower base is the pentagon ABCDE, terminates in an equal and parallel pentagon FGHIK, which is called the upper base. The side faces of the prism are the parallelograms DH, DK, EF,

K

F

AG, and BH. These are called the convex, or lateral surface of the prism

Of the Prism.

5. The altitude of a prism is the distance between its upper and lower bases: that is, it is a line drawn from a point of the upper base, perpendicular, to the lower base

6, A right prism is one in which the edges AF, BG, EK, HC, and DI, are perpendicular to the bases., In the right prism, either of the perpendicular edges is equal to the altitude. In the oblique prism the altitude is less than the edge.

K

F

H

E

7. A prism whose base is a triangle, is called a triangular prism; if the base is a quadrangle, it is called a quadrangular prism; if a pentagon, a pentagonal prism; if a hexagon a hexagonal prism; &c.

8. A prism whose base is a parallelogram, and all of whose faces are also parallelograms, is called a parallelopipedon. If all the faces are rectangles, it is called a rectangular parallelopipedon.

9. If the faces of the rectangular parallelopipedon are squares, the solid is called a cube: hence, the cube is a prism bounded by six equal squares

Of the Pyramid.

10. A pyramid is a solid, formed by several triangles united at the same point S, and terminating in the different sides of a polygon ABCDE.

The polygon ABCDE, is called the base of the pyramid; the point S, is called the vertex, and the triangles ASB, BSC, CSD, DSE, and ESA, form its lateral, or convex surface.

E

11. A pyramid whose base is a triangle, is called a trangular pyramid; if the base is a quadrangle, it is called a quadrangular pyramid; if a pentagon, it is called a petagonal pyramid; if the base is a hexagon, it is called a hexagonal pyramid; &c.

12. The altitude of a pyramid, is the perpendicular let fall from the vertex, upon the plane of the base. Thus, SO is the altitude of the pyramid S-ABCDE.

13. When the base of a pyramid is a regular polygon, and the perpendicular SO passes through the middle point of the base, the pyramid is called a right pyramid, and the line SO is called the axis