Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/260

236 Let AB,BC, two straight lines which meet one another, be parallel to two other straight lines DE, EF, which meet one another, but are not in the same plane with AB,BC: the plane passing through AB, BC, shall be parallel to the plane passing through DE, EF.

From the point B draw BG perpendicular to the plane passing through DE, EF, [XI. 11. and let it meet that plane at G; through G draw GH parallel to ED, and G parallel to EF. [1.31.

Then, because BG is perpendicular to the plane passing through DE, EF, [Construction.

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

but the straight lines GH and GK meet it, and are in that plane; therefore each of the angles BGH and BGK is a right angle.

Now because BA is parallel to ED, [Hypothesis. and GH is parallel to ED, [Construction. therefore BA is parallel to GH; [XI. 9. therefore the angles ABG and BGH are together equal to two right angles. [I. 29.

And the angle BGH has been shewn to be a right angle; therefore the angle ABG is a right angle. For the same reason the angle CBG is a right angle.

Then, because the straight line GB stands at right angles to the two straight lines BA, BC, at their point of intersection B, therefore GB is perpendicular to the plane passing through BA,BC. [XI. 4. And GB is also perpendicular to the plane passing through DE, EF. [Construction.