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 P Q be perpendicular to each

other, and at any point B of their intersection DQ let B A be drawn in the plane PQ, perpendicular

We are to prove A B I to the plane M N.
Draw BC in the plane MNI to D Q.
Then ZA B C is a right angle,
(being the plane 2 of the rt. dihedral Z formed by the two planes).

.. A B is I to the two straight lines DQ and BC.
. A B is I to the plane M N,

§ 449 (if a straight line be I to two straight lines drawn through its foot in a

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

Α

B

B Q
Fig. 1.

Fig. 2.
Let P Q (Fig. 1) be perpendicular to the plane MN, CQ

their intersection, and BA be drawn through any point B in C Q perpendicular to the plane MN.

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

At the point B draw B A' in the plane P Q I to the intersection CQ.

The line B A' will be I to the plane MN, § 472 (if two planes be I to each other, a straight line drawn in one of them I to

their intersection is I to the other). Now B A is I to the plane M N;

Hyp. .. BA and B A' coincide,

§ 447 (at a given point in a plane only one I can be erected to that plane). But B A' lies in the plane PQ; .. BA, which coincides with B A', lies in the plane P Q.

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.

PROPOSITION XIX. THEOREM. 475. If two intersecting planes be each perpendicular to a third plane, their intersection is also perpendicular to that plane.

Let the planes B D and B C intersecting in the line

A B be perpendicular to the plane PQ.

We are to prove

A B I to the plane P Q.

A perpendicular erected at B, a point common to the three planes, will lie in the two planes B C and BD,

§ 473 (if two planes be I to each other, a straight line drawn through any point

of intersection I to one of the planes will lie in the other plane).

And, since this I lies in both the planes, BC and B D, it must coincide with their intersection.

.. A B is I to the plane P Q.

Q. E. D.

476. COROLLARY. If a plane be perpendicular to each of two intersecting planes, it is perpendicular to the intersection of those planes.

PROPOSITION XX. THEOREM. 477. Every point in the plane which bisects a dihedral angle is equally distant from the faces of that angle.

Let plane A M bisect the dihedral angle formed by

the planes A D and AC; and let PE and PF be perpendiculars drawn from any point P in the plane A M to the planes A C and A D.

AD,

We are to prove PE=PF.

Through P E and P F pass a plane intersecting the planes AC and A D in 0 E and O F.

Join PO.
Now the plane P EF is I to each of the planes A C and

§ 471 (if a straight line be I to a plane, any plane embracing the line is I to that

plane); the plane P E F is I to their intersection A 0. $ 476 (If a plane be I to each of two intersecting plancs, it is I to the intersection

of these planes).

..ZPOE= Z POF, (being measures respectively of the equal dihedral & M-OA-C and M-OA-D).

..rt. A PO E =rt. A POF,

.PE=PF,
(being homologous sides of equal A).

$ 110

Q. E. D.

Q.E.D.

SUPPLEMENTARY PROPOSITIONS.

PROPOSITION XXI. THEOREM. 478. The acute angle which a straight line inakes with its own projection on a plane is the least angle which it makes with any line of that plane.

Let BA meet the plane M N at B, and let B A' be its

projection upon the plane M N, and B C any other
line drawn through B in the plane.
We are to prove ZA BA' < Z A B C.
Take

- BC=B A'.

Join A C.
In the A A B A and A BC,
A B = A B,

Iden.
BA' = BC,

Cons. but AA' < AC,

§ 448 la I is the shortest distance from a point to a plane). .. 2 A B A < 2 A B C,

§ 116 (if two sides of a Abe equal respectively to two sides of another, but the third

side of the first A be greater than the third side of the second, then the Zopposite the third side of the first A is greater than the opposite the third side of the second).

Q. E. D. EXERCISE. — The angle included by two perpendiculars drawn from any point within a dihedral angle to its faces, is the supplement of the dihedral angle.