Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/251

Rh PROPOSITION 6. THEOREM.

If two straight lines be at right angles to the same plane, they shall be parallel to one another.

Let the straight lines AB, CD be at right angles to the same plane: AB shall be parallel to CD.

Let them meet the plane at the points B,D; join BD; and in the plane draw DE at right angles to BD; [1. 11.

make DE equal to AB; [I. 3.

and join BE,AE,AD.

Then, because AB is perpendicular to the plane, [Hypothesis.

it makes right angles with every straight line meeting it in that plane. [XI. Def. 3.

But BD and BE meet AB, and are in that plane,

therefore each of the angles ABD, ABE is a right angle. For the same reason each of the angles CDB, CDE is a right angle.

And because AB 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 it has been shewn that BE is equal to DA;

therefore 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 EDA. [I 8.

But the angle ABE is a right angle,

therefore the angle EDA is a right angle,

that is, ED is at right angles to AD.