Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/248

224 PROPOSITION 3. THEOREM.

If two planes cut one another their common section is a straight line.

Let two planes AB, BC cut one another, and let BD be their common section: BD shall be a straight line.

If it be not, from B to D, draw in the plane AB the straight line BED, and in the plane BC the straight line BFD. [Postulate 1. Then the two straight lines BED, BFD have the same extremities, and therefore include a space between them; but this is impossible. [Axiom 10. Therefore BD, the common section of the planes AB and BC cannot but be a straight line.

Wherefore, if two planes &c.

PROPOSITION 4. THEOREM.

If a straight line stand at right angles to each of two straight lines at the point of their intersection, it shall also he at right angles to the plane which passes through them, that is, to the plane in which they are.

Let the straight line EF stand at right angles to each of the straight lines AB, CD, at E, the point of their intersection: EF shall also be at right angles to the plane passing through AB, CD.

Take the straight lines AE, EB, CE, ED, all equal to one another; join AD, CB; through E draw in the plane in which are AB, CD, any straight line cutting AD at G, and CB at H: and from any point F in EF draw FA, FG, FD, FC, FH, FB.