ON DIHEDRAL ANGLES.

470. DEF. The amount of rotation which one of two intersecting planes must make about their intersection in order to coincide with the other plane is called the Dihedral angle of the planes.

The Faces of a dihedral angle are the intersecting planes.

The Edge of a dihedral angle is the intersection of its faces. The Plane angle of a dihedral angle is the plane angle formed by two straight lines, one in each plane, perpendicular to the edge at the same point.

perpendicular to the edge A B at the same point P.

471. The plane angle of a dihedral angle has the same magnitude from whatever point in the edge we draw the perpendiculars. For every pair of such angles have their sides respectively parallel (§ 65), and hence are equal (§ 462).

A

Two equal dihedral angles, D-A B-C', and D-A B-E', have corresponding equal plane angles, DAC and DAE. This may be shown by superposition.

Any two dihedral angles, C-A B-E' and E-A B-H', have the same ratio as their corresponding plane angles, CA E and EA H. This may be shown by the method employed in $200 and $201.

Hence a dihedral angle is measured by its plane angle.

H

G

F

A

G

It must be observed that the sides of the plane angle which measures the dihedral angle must be perpendicular to the edge. Thus in the rectangular solid A H, Fig. 1, the

dihedral angle F-B A-H, is a right dihedral angle, and is measured by the angle CED, if its sides CE and ED, drawn in the planes AF and A G respectively, be perpendicular to A B. But angle C'E' D', drawn as represented in the diagram, is acute, while angle C" E" D", drawn as represented, is obtuse.

Many properties of dihedral angles can be established which are analogous to propositions relating to plane angles. Let the student prove the following:

1. If two planes intersect each other, their vertical dihedral angles are equal.

2. If a plane intersect two parallel planes, the exteriorinterior dihedral angles are equal; the alternate-interior dihedral angles are equal; the two interior dihedral angles on the same side of the secant plane are supplements of each other.

3. When two planes are cut by a third plane, if the exteriorinterior dihedral angles be equal, or the alternate dihedral angles be equal, or the two interior dihedral angles on the same side of the secant plane be supplements of each other, the two planes are parallel.

4. Two dihedral angles are equal if their faces be respectively parallel and lie in the same direction; or opposite directions, from the edges.

5. Two dihedral angles are supplements of each other if two of their faces be parallel and lie in the same direction, and the other faces be parallel and lie in the opposite direction, from the edges. rose to t the

PROPOSITION XVI. THEOREM.

472. If a straight line be perpendicular to a plane every plane embracing the line is perpendicular to that plane.

Let A B be perpendicular to the plane M N. We are to prove any plane, PQ, embracing AB, perpendicular to MN.

At B draw, in the plane MN, BCL to the intersection DQ. to MN, it is I to DQ and BC,

Since AB is

(if a straight line be

§ 430 to a plane, it is to every straight line in that plane drawn through its foot).

Now ABC is the measure of the

dihedral P-D Q-N.

But ABC is a right angle,

.. the P-D Q-N is a right dihedral,

.. PQ is L to MN.

§ 470

Q. E. D.

PROPOSITION XVII. THEOREM.

473. If two planes be perpendicular to each other, a straight line drawn in one of them perpendicular to their intersection is perpendicular to the other plane.

Let the planes M N and PQ be perpendicular to each other, and at any point B of their intersection DQ let BA be drawn in the plane PQ, perpendicular to DQ.

We are to prove A B to the plane M N.

Draw

Then

BC in the plane MNL to DQ.

ZABC is a right angle,

(being the plane of the rt. dihedral ▲ formed by the two planes).

.. A B is to the two straight lines DQ and B C.

.. A B is to the plane MN,

§ 449

(if a straight line be

to two straight lines drawn through its foot in a plane, it is to the plane).

Q. E. D.

474. If two planes be perpendicular to each other, a straight line drawn through any point of intersection perpendicular to one of the planes will lie in the other plane.

Let PQ (Fig. 1) be perpendicular to the plane MN, CQ their intersection, and BA be drawn through any point B in CQ perpendicular to the plane M N.

We are to prove that B A lies in the plane P Q.

At the point B draw B A' in the plane PQ to the inter

section C Q.

The line B A' will be to the plane MN,

§ 472

(if two planes be

to each other, a straight line drawn in one of them to their intersection is to the other).

Now

BA is to the plane MN;

.. BA and B A' coincide,

Hyp.

§ 447

(at a given point in a plane only one can be erected to that plane).

.. BA, which coincides with BA', lies in the plane PQ.

Q. E. D.

SCHOLIUM. Through a line parallel or oblique to a plane, as A C, Fig. 2, only one plane can be passed perpendicular to the given plane.