BOOK VI.

PLANES AND POLYEDRAL

ANGLES.

DEFINITIONS.

1. A straight line is PERPENDICULAR TO A PLANE, when it is perpendicular to every straight line of the plane which passes through its FOOT; that is, through the point in which it meets the plane.

In this case, the plane is also perpendicular to the line.

2. A straight line is PARALLEL TO A PLANE, when it cannot meet the plane, how far soever both may be produced. In this case, the plane is also parallel to the line.

3. TWO PLANES ARE PARALLEL, when they cannot meet, how far soever both may be produced.

4. A DIEDRAL ANGLE is the amount of divergence of two planes.

The line in which the planes meet, is called the edge of the angle, and the planes themselves are called faces of the angle.

The measure of a diedral angle is the same as that of a plane angle formed by two straight lines, one in each face, and both perpendicular to the edge at the same point. A diedral angle may be acute, obtuse, or a right angle. In the latter case, the faces are perpendicular to each other.

5. A POLYEDRAL ANGLE is the amount of divergence of several planes meeting at a common point.

This point is called the vertex of the angle; the lines in which the planes meet are called edges of the angle, and the portions of the planes lying between the edges are

called faces of the angle. Thus, S
is the vertex of the polyedral angle,
whose edges are SA, SB, SC,

SD,
BSC, CSD, DSA.

and whose faces are ASB,

A polyedral angle which has but

three faces, is called a triedral

S

D

angle.

POSTULATE.

A straight line may be drawn perpendicular to a plane from any point of the plane, or from any point without the plane.

PROPOSITION I. THEOREM.

If a straight line has two of its points in a plane, it will lie wholly in that plane.

For, by definition, a plane is a surface such, that if any two of its points be joined by a straight line, that line will lie wholly in the surface (B. I., D. 8).

Cor. Through any point of a plane, an infinite number of straight lines may be drawn which will lie in the plane. For, if a straight line be drawn from the given point to any other point of the plane, that line will lie wholly in the plane.

Scholium. If any two points of a plane be joined by a straight line, the plane may be turned about that line as an

axis, so as to take an infinite number of positions. Hence, we infer that an infinite number of planes may be passed through a given straight line.

PROPOSITION II. THEOREM.

Through three points, not in the same straight line, one plane can be passed, and only one.

Let A, B, and C be the three points: then can one plane be passed through them, and only one.

Join two of the points, as A and B, by the line AB. Through AB let a plane be passed, and let this plane be turned around AB until it contains the point ; in this position it will pass through the three points A, B, and C. If now, the plane be turned

B

about AB, in either direction, it will no longer contain the point C hence, one plane can always be passed through three points, and only one; which was to be proved.

Cor. 1. Three points, not in a straight line, determine the position of a plane, because only one plane can be passed through them.

Cor. 2. A straight line and a point without that line, determine the position of a plane, because only one plane can be passed through them.

Cor. 3. Two straight lines which intersect, determine the position of a plane. For, let AB and AC intersect at A: then will either line, as AB, and one point of the other, as C, determine the position of a plane.

.

Cor. 4. Two parallel straight lines determine the position of a

CD be parallel. By definition

plane. For, let AB and CD

(B. I., D. 16) two parallel lines always lie in the same plane. But either line, as AB, and any point of the other, as F, determine the position of a plane hence, two parallels determine the position of a plane.

A

B

C

D

F

PROPOSITION III. THEOREM.

The intersection of two planes is a straight line.

Let AB and CD be two planes: then will their intersection be a straight line.

For, let E and F be any two points common to the planes; draw

the straight line EF

ing two points in the

will lie wholly in that

This line hav

plane

AB,·

plane; and

having two points in the plane CD,

Ꭰ

will lie wholly in that plane: hence, every point of EF is common to both planes. Furthermore, the planes can have no common point lying without EF, otherwise there would be two planes passing through a straight line and a point lying without it, which is impossible (P. II., C. 2); hence, the intersection of the two planes is a straight line; which was to be proved.

If a straight line is perpendicular to two straight lines at their point of intersection, it is perpendicular to the plane of those lines.

Let MN be the plane of the two lines BB, CC, and let AP be perpendicular to these lines at P: then will

AP be perpendicular to every straight line of the plane which passes through P, and consequently, to the plane itself.

For, through P, draw in the plane MN, any line PQ; through any point of this line, as Q, draw the line BC, 80 that BQ shall be equal to QC (B. IV., Prob. V.); draw AB, AQ, and AC.

B

B

The base BC, of the triangle BPC, being bisected at Q, we have (B. IV., P. XIV.),

In like manner, we have, from the triangle ABC,

Subtracting the first of these equations from the second, member from member, we have,

AC2- PC2 + AB2 PB2 =

2A Q2 — `2PQ2.

But, from Proposition XI., C. 1, Book IV., we have,

The triangle APQ is, therefore, right-angled at P (B. IV., P. XIII., S.), and consequently, AP is perpendicular to PQ: hence, AP is perpendicular to every line of the plane MN passing through P, and consequently, to the plane itself; which was to be proved.