Elements of geometry and trigonometry . other, each of those lines is perpen-dicular to the plane of the other two, and the three planes areperpendicular to each other. PROPOSITION XVII. THEOREM. If tieo planes are perpendicular to each other, a line drawn inone of them perpendicuUw to their common intersection, willbe perpendicular to the oUier plajie. Let the plane AB be perpen-dicular| to NM ; then if the lineAP be perpendicular to the inter-section BC, it will also be perpen-dicular to the plane \M. For, in the plane AL\ draw PDperfxîndicular to PB ; then, be-cause the jilanes are perpendi
Elements of geometry and trigonometry . other, each of those lines is perpen-dicular to the plane of the other two, and the three planes areperpendicular to each other. PROPOSITION XVII. THEOREM. If tieo planes are perpendicular to each other, a line drawn inone of them perpendicuUw to their common intersection, willbe perpendicular to the oUier plajie. Let the plane AB be perpen-dicular| to NM ; then if the lineAP be perpendicular to the inter-section BC, it will also be perpen-dicular to the plane \M. For, in the plane AL\ draw PDperfxîndicular to PB ; then, be-cause the jilanes are perpendicu-lar, the angle APD is a right an-glf; ; therefore, the line AP isperp(Tidi(:ular to th(î two PB, PD ; therefore it is perpendicular to their plane MN(Prop. IV.). Cor. If the plane; AlJis perpendicular to the plane MN,an(Iif at a [K)int P ot the coiimion we erect a perpen-dicular to the- plan(î M \, perjundicuhir uill be in the planoAB; for, if not, then, in the plane AB we nnghldraw AP per-M- 18. ns GEOMETRY. pendicular lo PB the common intersection, and this AP, at thesame time, would he perpendicular to the plane MN; thereforeat the same point P there would be two perpendiculars to theplane MN, wiiicii is impossible (Prop. IV» Cor. 2.). PEOPOSÏTION XVni. THEOREM. Iftwoplawes are ])erpendicuîa7 to a third plane, their commonintersection will also be perpendicular to tht third plane. Let the planes AB, AD, be per-pendicular to NM; then will theirintersection AP be perpendicularto NM. For, at the point P, erect a per-pendicular to the plane MN ; thatperpendicular must be at once inthe plane AB and in tlie plane Al>(Prop. XVII. Cor.) ; therefore itis theii common intersection AP. M w D B- y E ijNT PROPOSITION XIX. THEOREM. If a solid angle is formed by three plane angles, the sum of anytwo of these angles will be greater than the third. The proposition requires demonstra-tion only when the plane angle, whichis compared to the sum of
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