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 interfects them, the angles BGH, GBA 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 also 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 also 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.

PRO P. XII. THEOREM.

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

H

B

Let the two parallel planes AB, CD be cut by the plane EGHF; then will their common fections EF, GH be parallel 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 alfo be in that plane.

Since, therefore, the point K 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 fame time, which is abfurd.

The lines EF, GH, therefore, do not meet on the side FH; and, in the fame manner, it may be proved, that they do not meet on the fide EG; confequently they are parallel to each other. Q. E. D.

PROP. XIII. THEOREM.

If a right line be perpendicular to a plane, every plane which passes through it will also be perpendicular to that plane.

A H

F B

Let the right line AB be perpendicular to the plane CK; then will every plane which paffes through that line be also perpendicular to CK.

For let ED be any plane which paffes by the line AB; and in this plane draw any right line GF perpendicular to the common fection CE (I. 11.)

Then, because the line AB is perpendicular to the plane CK (by Hyp.), it will also be perpendicular to the line CE; and the angle ABF will be a right angle (VII. Def. 2.)

And fince the angles ABF, GFB are each of them a right angle, and the lines AB, GF are in the fame plane, they will be parallel to each other (VII. 4.)

Since, therefore, thefe lines are parallel to each other, and one of them, AB, is perpendicular to the plane CK,

the other, GF, will also be perpendicular to that plane (VII. 5.)

But one plane is perpendicular to another, when any right line that can be drawn in it, at right angles to the common fection, is also at right angles to the other plane (VII. Def. 3.); whence the plane ED is perpendicular to the plane CK, as was to be fhewn.

PROP. XIV. THEOREM.

If two planes which cut each other, be each of them perpendicular to a third plane, their common fection will also be perpendicular to that plane.

E

D

Let the two planes AB, CB be each of them perpendicular to the plane ACD; then will their common fection ED be alfo perpendicular to ACD.

For if not, let DE be drawn in the plane AB, at right angles to the common fection AD; and DF in the plane CB at right angles to the common fection DC (I. 11.)

Then because the plane AB is perpendicular to the plane ACD (by Hyp.), the line DE will also be perpendi. cular to that plane (VII. 3.)

And fince the plane CB is perpendicular to the plane ACD (by Hyp.), the line DF will also be perpendicular to that plane (VII. 3.)

But lines which are perpendicular to the fame plane are parallel to each other (VII. 4.); whence the lines DE, DF meet, and are parallel at the fame time, which is abfurd.

Thefe lines, therefore, are not perpendicular to the plane ACD; and the fame may be fhewn of any other line but DB; whence DB is perpendicular to ACD, as was to be fhewn.