Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/61

Rh PROPOSITION 33. THEOREM.

The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel.

Let AB and CD be equal and parallel straight lines, and let them be joined towards the same parts by the straight lines AC and BD : AC and BD shall be equal and parallel.

Join BC.

Then because AB is par- allel to CD, [Hypothesis. and BC meets them, the alternate angles ABC, BCD are equal. [I. 29. And because AB is equal to CD, [Hypothesis. and BC is common to the two triangles ABC, DCB ; the two sides AB, BC are equal to the two sides DC, CB, each to each ; and the angle ABC was shewn to be equal to the angle BCD; therefore the base AC is equal to the base BD, and the triangle ABC to the triangle BCD, and the other angles to the other angles, each to each, to which the equal sides are opposite ; [I. 4. therefore the angle ACB is equal to the angle CBD.

And because the straight line BC meets the two straight lines AC, BD, and makes the alternate angles ACB, CBD equal to one another, AC is parallel to BD, [I. 27. And it was shewn to be equal to it.

Wherefore, the straight lines &c.

PROPOSITION 34. THEOREM.

The opposite sides and angles of a parallelogram are equal to one another, and the diameter bisects the par- allelogram, that is, divides it into two equal parts.

Note. A parallelogram is a four-sided figure of which the opposite sides are parallel ; and a diameter is the straight line joining two of its opposite angles.