If two straight lines are parallel, and one of them is at right angles to any plane, then the remaining one is also at right angles to the same plane.

Let *AB* and *CD* be two parallel straight lines, and let one of them, *AB,* be at right angles to the plane of reference.

I say that the remaining one, *CD,* is also
at right angles to the same plane.

Let *AB* and *CD* meet the plane of reference at the points *B* and *D.* Join *BD.* Then *AB, CD,* and *BD* lie in one plane.

Draw *DE* in the plane of reference at right angles to *BD,* make *DE* equal to *AB,* and join *BE, AE,* and *AD*

Now, since *AB* is at right angles to the plane of reference, therefore *AB* is also at right angles to all the straight lines which meet it and lie in the plane of reference. Therefore each of the angles *ABD* and *ABE* is right.

And, since the straight line *BD* falls on the parallels *AB* and *CD,* therefore the sum of the angles *ABD* and *CDB* equals two right angles.

But the angle *ABD* is right, therefore the angle *CDB* is also right. Therefore *CD* is at right angles to *BC.*

And since *AB* equals *DE,* and *BD* is common, the two sides *AB* and *BD* equal the two sides *ED* and *DB,* and the angle *ABD* equals the angle *EDB,* for each is right, therefore the base *AD* equals the base *BE.*

And since *AB* equals *DE,* and *BE* equals *AD,* the two sides *AB* and *BE* equal the two sides *ED* and *DA* respectively, and *AE* is their common base, therefore the angle *ABE* equals the angle *EDA.*

But the angle *ABE* is right, therefore the angle *EDA* is also right. Therefore *ED* is at right angles to *AD.* But it is also at right angles to *DB.* Therefore *ED* is also at right angles to the plane through *BD* and *DA.*

Therefore *ED* also makes right angles with all the straight lines which meet it and lie in the plane through *BD* and *DA.* But *DC* lies in the plane through *BD* and *DA* inasmuch as *AB* and *BD* lie in the plane through *BD* and *DA,* and *DC* also lies in the plane in which *AB* and *BD* lie.

Therefore *ED* is at right angles to *DC,* so that *CD* is also at right angles to *DE.* But *CD* is also at right angles to *BD.* Therefore *CD* is set up at right angles to the two straight lines *DE* and *DB* so that *CD* is also at right angles to the plane through *DE* and *DB.*

But the plane through *DE* and *DB* is the plane of reference, therefore *CD* is at right angles to the plane of reference.

Therefore, *if two straight lines are parallel, and one of them is at right angles to any plane, then the remaining one is also at right angles to the same plane.*

Q.E.D.

This proposition is used in the proof of the next one as well as several others in this book.