PROPOSITION 8. THEOREM.
If two straight lines he parallel, and one of them be at right angles to a plane, the other also shall he at right angles to the same plane.
Let AB, CD be two parallel straight lines; and let one of them AB be at right angles to a plane: the other CD shall be at right angles to the same plane.
Let AB, CD meet the plane at the points B, D; join BD; therefore AB, CD, BD are in one plane. [XI. 7.
In the plane to which AB is at right angles, draw DE at right angles to BD; [I. 11.
make DE equal to AB; [I 3.
and join BE,AE,AD.
Then, because AB is at right angles to the plane, [Hypothesis.
it makes right angles with every straight line meeting it in that plane; [XI. Definition 3.
therefore each of the angles ABD, ABE is a right angle.
And because the straight line BD meets the parallel straight lines AB, CD,
the angles ABD, CDB are together equal to two right angles, [I. 29.
But the angle ABD is a right angle, [Hypothesis. therefore the angle CDB is a right angle;
that is, CD is at right angles to BD.
And because is equal to ED, [Construction.
and BD is common to the two triangles ABD, EDB;
the two sides AB, BD are equal to the two sides ED, DB, each to each;
and the angle ABD is equal to the angle EDB, each of them being a right angle; [Axiom 11.
therefore the base AD is equal to the base EB. [I. 4.
Again, because AB is equal to ED, [Construction.
and BE has been shewn equal to DA,
