Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/253

Rh 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,