Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/258

234 PROPOSITION 13. THEOREM.

From the same point in a given plane, there cannot be two straight lines at right angles to the plane, on the same side of it; and there can he hut one perpendicular to a plane from, a point without the plane.

For, if it be possible, let the two straight lines AB,AC be at right angles to a given plane, from the same point A in the plane, and on the same side of it.

Let a plane pass through BA, AC; the common section of this with the given plane is a straight line; [XI. 3. let this straight line be DAE.

Then the three straight lines AB, AC, DAE are all in one plane. And because CA is at right angles to the plane, [Hypothesis. it makes right angles with every straight line meeting it in the plane. [XI. Definition 3. But DAE meets CA, and is in that plane; therefore the angle CAE is a right angle. For the same reason the angle BAE is a right angle. Therefore the angle CAE is equal to the angle BAE; [Ax.ll. and they are in one plane; which is impossible. [Axiom 9.

Also, from a point without the plane, there can be but one perpendicular to the plane.

For if there could be two, they would be parallel to one another, [XI. 6. which is absurd.

Wherefore, from the same point &c.