AG (I. 11.), and it will also be perpendicular to the plane BC, as was required..

For, fince the right line FA is perpendicular to each of the right lines AE, AG (by Conft.), it will also be perpendicular to the plane EG which passes through those lines (VII. 3.)

And because a right line which is perpendicular to a plane is perpendicular to every right line which meets it in that plane (Def. 2.), FA will be perpendicular to AH.

But AG is also perpendicular to AH (by Conft.); whence AH, being perpendicular to each of the right lines FA, AG, it will also be perpendicular to the plane BC (VII. 3.), as was to be fhewn.

PROP. IX. PROBLEM.

To draw a right line perpendicular to a given plane, from a given point above it.

H

Let A be the given point, and BG the given plane; it is required to draw a right line from the point A that fhall be perpendicular to the plane BG.

Take any right line BC, in the plane BG, and draw AD perpendicular to BC (I. 11.); then if it be alfo perpendicular to the plane BG, the thing required is done.

But if not, draw DE, in the plane BC, at right angles to BC (I. 11.), and make AF perpendicular to DE (I. 12.);

then will AF be perpendicular to the plane BG, as was required.

For, through the point F, draw the line HG parallel to the line BC (I. 27.)

Then fince the right line BC is perpendicular to each of the right lines DA, DE, it will also be perpendicular to the plane which paffes through thofe lines (VII. 3.)

And because the lines BC, HG are parallel to each other, and one of them, BC, is perpendicular to the plane ADF, the other, HG, will also be perpendicular to that plane (VII. 5.)

But if a line be perpendicular to a plane it will be perpendicular to all the lines which meet it in that plane (VII. Def. 2.); whence the line HG is perpendicular to

AF.

And fince the line AF is perpendicular to each of the lines HG, ED, at their point of interfection F, it will also be perpendicular to the plane BG (VII. 3.), as was to be fhewn.

PROP. X. THEOREM.

Planes to which the fame right line is perpendicular, are parallel to each other.

A

Let the right line AB be perpendicular to each of the planes CD, EF; then will those planes be parallel to each other.

For

For if they be not, let them be produced till they meet each other; and in the line GH, which is their common fection, take any point K; and join KA, KB:

Then, because the line AB is perpendicular to the plane EF (by Hyp.), it will also be perpendicular to the line BK, which lies in that plane (VII. Def. 2.); and the angle ABK will be a right angle.

And, for the fame reason, the line AB, which is perpendicular to the plane DC (by Hyp.), will be perpendicular to the line AK; and the angle BAK will also be a right angle.

The angles ABK, BAK are, therefore, equal to two right angles, which is abfurd (1. 28.); and confequently the planes can never meet, but must be parallel to each other (VII. Def. 6.), as was to be fhewn.

PRO P. XI. THEOREM.

If two right lines which meet each other, be parallel to two other right lines which meet each other, though not in the fame. plane with them, the planes which pass through those lines will be parallel.

Let the right lines AB, BC, which meet each other in B, be parallel to the right lines DE, EF, which meet each ether in E, though not in the fame plane with them; then will the plane ABC be parallel to the plane DEF.

For through the point в draw BG perpendicular to the plane DFE (VII. 9.); and make GH parallel to DE, and GK to EF (I. 27.)

Then because BG is at right angles with the plane DFE, it will also be at right angles with each of the lines GH, GK which meet it in that plane (Def. 2.)

And fince GH is parallel to DE or AB (by Conft, and VII. 6.), and BG intersects them, the angles BGH, GBÁ are, together, equal to two right angles (I. 25.)

But the angle BGH has been fhewn to be a right angle; whence the angle GBA is alfo a right angle; and confequently GB is perpendicular to BA,

And, in the fame manner, it may be fhewn, that GB is perpendicular to BC.

The right line GB, therefore, being perpendicular to each of the right lines BA, BC, will alfo be perpendicular to the plane ACB through which they pass (VII. 3.)

But planes to which the fame right line is perpendicular are parallel to each other (VII. 10.); whence the plane ACB is parallel to the plane DFE

PROP. XII. THEOREM.

If any two parallel planes be cut by ano, ther plane, their common fections will be parallel.

Let the two parallel planes AB, CD be cut by the plane GHF; then will their common fections EF, GH be pa. rallel to each other.

For if EF, GH be not parallel, they may be produced till they meet, either on the fide FH, or the fide EG.

Let them be produced on the fide FH, and meet each other in the point K.

Then, fince the whole line EFK is in the plane AB, or the plane produced, the point K must be in that plane.

And because the whole line GHK is in the plane CD, or the plane produced, the point K muft also be in that plane.

Since, therefore, the point x is in each of the planes AB, CD, those planes, if produced, will meet in that point.

But the two planes are parallel to each other, by hypothefis; whence they meet, and are parallel, at the same time, which is abfurd.