Conversely: If two straight lines be parallel, and one of them be at right angles to a plane, the other shall also be at right angles to the same plane. Let AB be to CF, and CF be 1 to the plane BFG. Join BF, and in the plane BFG draw GF1 to BF and = AB. Join BG, AG, AF. Then . CF is 1 plane BFG, .. LS CFB, CFG are rights; and AB is I to CF, .. LS ABF, CFB are together two right s;.. ABF is a right 4, = (I. 21) and GF is 1 to CF, and also 1 to BF; .. GF is 1 to the plane in which are CF, BF (Prop. 6), and .. 1 to AF, .. LAFG is a right . Again, AB, BF and ▲ ABF are respectively = GF, FB and GFB; .. AF = GB. (I. I) Hence AF, FG, GA are respectively GB, BA, AG; = (1.5) and also LABF is a right 4; .. LABG is a right 4, .. AB is to the plane BFG. (Prop PROPOSITION VIII. It is always possible to draw a straight line perpendicular to a plane from a given point without it. K For let AB be a given plane, and P a point without it, and let it be required to draw from P a straight line 1 to the plane AB. In the plane AB draw any straight line CF, and from Plet fall PG 1 to CF. Then if PG is also to the plane AB, the thing required has been done. But, if not, in the plane AB draw GQ1 to CF, and from P let fall PQ1 to GQ; then PQ is 1 to the plane AB. Through Q draw HK || to CF. Then. CF is 1 PG and also 1 QG; ... CF is to plane PGQ. (Prop. 6) But HK is and is.. to CF, ... HK is 1 to plane PGQ, (Prop. 7) to PQ; .. PQ is to HK, and PQ is also to QG; .. PQ is to plane AB. (Prop. 6) PROPOSITION IX. There cannot be drawn more than one straight line perpendicular to a plane from a given point without it. For, if possible, let PQ, PR be each of them to the plane AB. Join QR. Then PQR, PRQ are each of them right angles; which is impossible. (1. 13) PROPOSITION X. It is always possible to draw a straight line perpendicular to a plane from a given point within it. For let AB be a given plane, and C a point within it, and let it be required to draw from C a straight line 1 to the plane AB. From any point P without the plane let PQ be drawn 1 to the plane. Through C draw CF || to PQ. Then shall CF be 1 to the plane AB. For if two straight lines are I and one of them is to a plane, the other is also to the same plane. (Prop. 7) PROPOSITION XI. There cannot be drawn more than one straight line perpendicular to a plane from a given point within it. A F For, if it be possible, from a point C within the plane AB, let two straight lines CF, CG be drawn to the plane AB, and let the plane through CF, CG cut the plane AB in XY. Then s FCY, GCY are each rights, and are .. equal to one another; which is impossible. |