the two sides AB, BE are equal to the two sides ED DA, each to each;
and the base AE is common to the two triangles ABE, EDA;
therefore the angle ABE is equal to the angle ADE. [I. 8.
But the angle ABE is a right angle;
therefore the angle ADE is a right angle;
that is, ED is at right angles to AD.
But ED is at right angles to BD, [Const.
therefore ED is at right angles to the plane which passes through BD, DA, [XI. 4.
and therefore makes right angles with every straight line meeting it in that plane. [XI. Definition 3.
But CD is in the plane passing through BD, DA, because all three are in the plane in which are the parallels AB, CD;
therefore ED is at right angles to CD,
and therefore CD is at right angles to ED.
But CD was shewn to be at right angles to BD;
therefore CD is at right angles to the two straight lines BD, ED, at the point of their intersection D,
and is therefore at right angles to the plane passing through BD, ED, [XI. 4,
that is, to the plane to which AB is at right angles.
Wherefore, if two straight lines &c. q.e.d.
PROPOSITION 9. THEOREM.
Two straight lines which are each of them parallel to the same straight line, and not in the same plane with it, are parallel to one another.
Let AB and CD be each of them parallel to EF, and not in the same plane with it: AB shall be parallel to CD.
In EF take any point G; in the plane passing through EF and AB, draw from G the straight line GH at right angles to EF;
and in the plane passing through EF and CD, draw from G the
