1. Because the plane AC is perpendicular to MN, the two lines OD, OP, one in each plane at right angles to the line of intersection, form a right angle (§ 623).

2. Because OD is perpendicular to AB (hyp.) and to OP (1), it is perpendicular to the plane of these lines; that is, to the plane AC. Q.E.D.

THEOREM XXVII.

631. If two planes be perpendicular to each other, every line perpendicular to the one is either parallel to the other or lies in the other.

Hypothesis. MN, any plane; ABCD, a plane perpendicu

1. Because RQ is perpendicular to AB and lies in the plane MN, it is perpendicular to the plane ABCD (§ 630).

2. Because the lines OP and QR are each perpendicular to the plane ABCD, they are parallel (§ 590).

3. Because the plane MN contains one of these parallels, QR, it is either parallel to the other, OP, or contains it (§ 607). Q.E.D.

Corollary. The plane MN will contain the perpendicular OP when O lies on the line of intersection AB. Hence:

632. If two planes be perpendicular, a line through their common intersection perpendicular to the one will lie in the other.

THEOREM XXVIII.

633. If each of two planes is perpendicular to a third plane, their line of intersection is also perpendicular to that third plane.

Hypothesis. PQ, RS, two planes, each perpendicular to plane MN; AB, their

R

N

B on their line of intersection a perpendicular to MN, it will lie in the plane PQ (§ 632).

2. Because RS 1 MN, this line will also lie in RS.

3. Therefore the line will be the line of intersection of the two planes, or AB; whence AB is the perpendicular to the plane MN from the point B. Q.E.D.

Corollary. Because a plane perpendicular to two planes is perpendicular to their line of intersection, and because all planes perpendicular to the same line are parallel or coincident (§ 616), we conclude:

634. All planes perpendicular to the same two planes are either parallel or coincident.

THEOREM XXIX.

635. If two planes are respectively perpendicular to two intersecting lines, their line of intersection is perpendicular to the plane of the lines.

Hypothesis. OH, OI, two lines intersecting at 0;
Plane MN line OH;

Plane KL

line OI;

H

M

מי

K

N

L

($629)

Plane MN plane HOI.

2. In the same way,

Plane KL plane HOI.

3. Because each of the planes is perpendicular to HOI and UV is their line of intersection,

UV plane HOI (§ 633). Q.E.D.

Relations of Three or more Planes.

636. REMARK. When three planes, which we may call X, Y, and Z, mutually intersect, there will be three lines of intersection:

One line formed by the planes X and Y;

One line formed by Y and Z;

One line formed by Z and X.

THEOREM XXX.

637. The three lines of intersection of three planes are either parallel or meet in a point.

Proof. Let us call the three planes X, Y, and Z. Let us also call

Then

a, the line of intersection of X and Y;
b, the line of intersection of Y and Z;
c, the line of intersection of Z and X.

1. Because the lines a and b both lie in the plane Y they are either parallel or intersect each other. The same may be shown for b and c, and for c and a.

2. Suppose a and b to intersect. Because a lies in both the planes X and Y, and b lies in both Y and Z, the point where they intersect must lie in all three planes X, Y, and Z. Therefore it must lie on both the planes X and Z, and therefore on their line of intersection c. The three lines a, b, and c will then all meet at this point.

3. If a and b are parallel, c cannot meet either of them, because, by (2), where it meets the one it must meet the other also. Therefore, in this case, none of the lines will ever meet any of the others, and because each pair lies in the same plane they must be all parallel.

638. Corollary 1. If the three lines of intersection of three planes meet in a point, the three planes all pass through that point.

639. Cor. 2. If two lines are parallel to each other, each of them is also parallel to the intersection of any two planes, one of which passes through each of the lines.

THEOREM XXXI.

640. If the lines of intersection of three planes are parallel, any fourth plane perpendicular to two of the three planes is also perpendicular to the third. Hypothesis. The parallel lines a, b, c are the lines of intersection of three planes,

plane MN is perpendicular to both the planes ab and bc, it is perpendicular to their line of intersection b (§ 633).

2. Because MN is perpendicular to b, it is perpendicular to the lines a and c, parallel to b (§ 591).

3. Therefore it is perpendicular to the plane ac, which passes through those lines (§ 629). Q.E.D.

CHAPTER !!!.

OF POLYHEDRAL ANGLES.

641. Def. When three or more planes pass through the same point, they are said to form a polyhedral angle at that point.

A polyhedral angle is also called a solid angle.

Each plane which forms a polyhedral angle is supposed to be cut off along its lines of intersection with the planes adjoining it on each side.

642. Def. Edges of a polyhedral angle are the straight lines along which the planes intersect.

D

B

A polyhedral angle. O, the vertex; OA, OB, OC, etc., the edges; AOB, BOC,

643. Def. Faces of a poly- etc., the faces. hedral angle are the planes which form it.

644, Def. The vertex of a polyhedral angle is the point where the faces and edges all meet.

645. The edges of a polyhedral angle may be produced indefinitely. But to make the study of the angle easy, the faces and edges may be supposed cut off by a plane. The intersection of the faces with this plane will then form a polygon, as ABCDE.

This polygon is the base of the polyhedral angle.

646. Each pair of faces which meet an edge form a dihedral angle along that edge. There are as many edges as faces, and therefore as many dihedral angles as faces.

Hence two classes of angles enter into any polyhedral angle, namely:

I. The plane angles AOB, BOC, COD, etc., called also face angles, which the edges form with each other. The planes on which these angles are measured are the faces.