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. .. A B is I to the two straight lines DQ and BC. § 449 (if a straight line be I to two straight lines drawn through its foot in a plane, it is I to the plane). Q. E. D. 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. 2. 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. § 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, $ 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 - BC=B A'. Join A C. Iden. 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. |