CHAPTER XXX.

DIHEDRAL ANGLES.

420. Dihedral Angles.

DEFINITION.Two half-planes drawn from the same line are said to form a dihedral angle. (See Fig. 551.)

Two intersecting complete planes accordingly form four dihedral angles (see Fig. 518), just as two intersecting straight lines form four angles.

Measure of a Dihedral Angle.-A dihedral angle is measured by the angle formed by the two lines (one in each half-plane) drawn from any point in the intersection of the half-planes perpendicular to the line of intersection.

DEFINITION. Two intersecting planes are said to be perpendicular if any one of the four dihedral angles which they form is a right angle.

It is easily proved that if one of the four dihedral angles is a right angle, each of the others is also a right angle.

421. THEOREM.-If a straight line is perpendicular to a given plane then any plane passing through this straight line is also perpendicular to the given plane.

In Fig. 552, AB I plane X :

Required to prove plane Y 1 plane X.

Cons.--Let AC be the intersection of the planes X, Y. Draw AD in plane X L AC.

But

plane X, ..

FIG. 553.

BAC BAD R.

= L

AB, AD are each 1 AC,

..the dihedral angle is measured by

=

BAD.

BAD = R; hence plane Y plane X.

422. THEOREM.-If two planes are perpendicular any line in one plane perpendicular to their intersection is perpendicular to the other plane.

In Fig. 552, plane Y plane X, and AB 1 AC.
Required to prove AB 1 plane X.

Cons. In plane X draw AD 1 AC.

423. THEOREM.-If two intersecting planes are each perpendicular to a third, then the line of intersection of these two planes is perpendicular to the third.

In Fig. 553, planes Y, Z are each 1 plane X :-
Required to prove that NP plane X

Cons.-Let NQ, NR be the intersections of planes Y

and Z with plane X.

In plane X draw NA 1 NQ, and NB NR.

Proof. plane X 1 plane Y, and. NA (in plane X)

Hence

intersection NQ,

NA plane Y.

§ 422.

ΝΑ Γ ΝΡ.

Similarly NB plane Z, whence NB NP.

· NPL NA, and NP 1 NB,
.. NP 1 plane X.

Proved.

§ 411.

424. PROBLEM.-Through any given straight line to draw a plane perpendicular to a given plane.

CASE I.-If given straight line 1 given plane, then any

plane through given straight line 1 given plane.

CASE II.—If the given straight

line AB is not given plane X

Cons. From A draw AC L

(Fig. 554).

plane X.

Then plane BAC is 1 plane X. Proof. AC (in plane BAC) plane X,

.. plane BAC plane X. § 421.

COROLLARY.-The projection of a given straight line on a given plane is a straight line, unless the given straight line is perpendicular to the plane.

N

B

D

§ 421.

X

FIG. 554.

In Fig. 554, the projection of AB on X is the line CD, which is the intersection of planes BAC and X.

Proof. Let P be any point on AB; draw PN 1 CD.
Then since plane BACD plane X, and PN 1 CD,

.. PN plane X.

§ 422.

Thus the projection of P on plane X is a point N on CD; and similarly for any other point on AB. Hence the projection of AB on X is CD,

SOL. GEO.

N N

425. THEOREM.--If a straight line intersects a plane, then the acute angle which the straight line makes with its projection on the plane is less than the angle which it makes with any other straight line in the plane which meets it.

In Fig. 555, AB meets plane X at A; BC 1 plane X (so that AC is projection of AB); and AD is any other line drawn from A in plane X.

Required to prove

BAD BAC.

Proof.-(i) If BAD is obtuse or a right angle it is

necessarily greater than BAC.

(ii) If

BAD is acute, draw BE AD. Bisect AB at F,

.. F is centre of the two Os AEB, ACB.

In As BFE, BFC, FB FB, FE FC, and BE > BC;

=

The inclination of a given straight line to a given plane is measured by the acute angle between the given straight line and its projection on the given plane (see Fig. 556).

The inclination of two given lines (of given senses) which do not meet and are not parallel is measured by the angle between two lines drawn from any point respectively parallel to the two given lines and in the

same senses.

If two different points are taken it is easily proved that the angles obtained are the same in each case. For the two lines drawn from the one point being respectively parallel to the two given lines are respectively parallel to the two lines drawn from the other point (§ 400). Hence the angle contained by the one pair of lines is equal to the angle contained by the other (§ 401).

FIG. 556.

THEOREM.-If a straight line is perpendicular to two intersecting lines which it does not meet, it is perpendicular to the plane containing them.

Prove by SS 411, 416.

COROLLARY.-If a straight line is perpendicular to two sides of a triangle it is also perpendicular to the third.

EXERCISES CXXXIX.

1. Show that if two dihedral angles are congruent, their measures are equal; and conversely.

2. If two planes intersect prove that two vertically opposite dihedral angles are equal and two adjacent dihedral angles are supplementary. 3. If a plane intersects two parallel planes prove that any two of the dihedral angles so formed are either equal or supplementary.

4. If three planes intersect in three parallel lines show that the sum of the three interior dihedral angles is equal to two right angles.

5. Show how to bisect a given dihedral angle.

6. Equal straight lines drawn from the same point to a given plane are equally inclined to the plane.

7. Prove § 423 indirectly by drawing lines from W in planes Y and Z perpendicular to their intersections with plane X.