Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/256

232 and AD is also equal to CF, [Axiom 1.

Therefore AC is equal and parallel to DF. [I. 33.

And because AB, BC are equal to DE, EF, each to each, and the base AC is equal to the base DF, therefore the angle ABC is equal to the angle DEF. [I. 8.

Wherefore, if two straight lines &c.

PROPOSITION 11. PROBLEM.

To draw a straight line perpendicular to a given plane from a given point without it.

Let A be the given point without the plane BH: it is required to draw from the point A a straight line perpen- dicular to the plane BH.

Draw any straight line BC in the plane BH, and from the point A draw AD perpendicular to BC. [1. 12. Then if AD be also perpendicular to the plane BH, the thing required is done. But, if not, from the point D draw, in the plane BH, the straight line DE at right angles to BC, [I. 11. and from the point A draw AF perpendicular to DE. [1. 12. AF shall be perpendicular to the plane BH.

Through F draw GH parallel to BC. [I. 31. Then, because BC is at right angles to ED and DA, [Constr. BC is at right angles to the plane passing through ED and DA. [XI. 4. And GH is parallel to BC; [Construction.