Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/247

Rh Then since the straight line AB is in the plane, it can be produced in that plane; let it be produced to D; and let any plane pass through the straight line,and be turned about until it pass through the point C. Then, because the points B and C are in this plane, the straight line BC is in it. [I. Definition 7. Therefore there are two straight lines ABC, ABD in the same plane, that have a common segment AB; but this is impossible. [I. 11, Corollary.

Wherefore, one part of a straight line &c.

PROPOSITION 2. THEOREM.

Two straight lines which cut one another are in one plane; and three straight lines which meet one another are in one plane.

Let the two straight lines AB, CD cut one another at E: AB and CD shall be in one plane; and the three straight lines EC, CB, BE which meet one another, shall be in one plane.

Let any plane pass through the straight line EB, and let the plane be turned about EB, produced if necessary, until it pass through the point C.

Then, because the points E and C are in this plane, the straight line EC is in it; [I. Definition 7. for the same reason, the straight line BC is in the same plane; and, by hypothesis, EB is in it. Therefore the three straight lines EC, CB, BE are in one plane. But AB and CD are in the plane in which EB and EC are; [XI. 1. therefore AB and CD are in one plane.

Wherefore, two straight lines &c.