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 A B be |to CF, and CF be 1 to the plane BFG. Then shall AB be I to the plane BFG. PROPOSITION VIII. It is always possible to draw a straight line perpendicular to a plane from a given point without it. 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 P let fall PG 1 to CF. Then if PG is also I to the plane AB, the thing required has been done. But, if not, in the plane AB draw GQ I to CF, and from P let fall PQ 1 to GQ; then PQ is 1 to the plane AB. 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 I to the plane AB. Join QR. Then PQR, PRQ are each of them right angles ; (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 I to the plane. Through C draw CF || to PQ. For if two straight lines are || and one of them is 1 to a plane, the other is also I 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. For, if it be possible, from a point C within the plane AB, let two straight lines CF, CG be drawn I to the plane AB, and let the plane through CF, CG cut the plane AB in XY. Then zs FCY, GCY are each right zs, and are ... equal to one another; which is impossible. |