THEOREM IV.

Every plane which contains the normal to another plane is perpendicular to that plane.

Let AB be normal to the plane MN; it is required to prove that any plane ABC which contains AB is perpendicular to MN.

Proof. Let BD in MN be a perpendicular to BC, common section of the two planes.

the

Because AB is perpendicular to the plane MN it is perpendicular to BC and BD, .... (Def. 5).

But the angle between the planes is measured by the angle ABD, (Def. 6).

......

Therefore the plane ABC is perpendicular to MN, ...... (Def. 7).

THEOREM V.

If two planes be perpendicular to one another, every line drawn in one of them perpendicular to their common section shall be perpendicular to the other.

Let the plane AC be perpendicular to the plane MN, and

the line AB in AC perpendicular to BC, the common section of the two planes; it is required to prove that AB is a normal to the plane MN.

MN.

Proof.

Let BD be a perpendicular to BC in the plane

Then because the plane AC is perpendicular to MN, the angle ABD is a right angle, (Def. 7).

......

Therefore AB is perpendicular to BD and BC, and consequently to the plane DC, ...... (Theor. III.).

Cor. If from any point of the plane AC a normal be drawn to MN, that normal must lie wholly in AC, for if not, two normals could be drawn to MN from the same point in AC, which is impossible, ...... (Theor. III. Cor. 3).

THEOREM VI.

If two planes which cut one another be both perpendicular to a third plane, their common section shall be perpendicular to the same plane.

Let the planes AC and AD be each perpendicular to MN; it is required to prove that their common section AB is perpendicular to MN.

Proof. Let BC and BD be the lines of intersection of the planes AC and AD with MN.

The line drawn through B perpendicular to the plane MN must lie wholly in the plane AC,...... (Theor. v. Cor.). Similarly it must lie in AD.

Therefore AB is the normal to MN at the point B.

THEOREM VII.

Two straight lines which are perpendicular to the same plane are parallel to one another.

Let the straight lines AB and CD be each perpendicular to MN; it is required to prove that they are parallel to one another.

Proof. Let B and D be the points of intersection of the lines with MN.

Because AB is perpendicular to MN the plane ABD is perpendicular to it also, ...... (Theor. IV.).

Because the plane ABD is perpendicular to MN, and the line DC is drawn through D perpendicular to MN, it lies wholly in the plane ABD, (Theor. v. Cor.).

Therefore AB and CD are in the same plane.

Also, since AB and CD are each perpendicular to MN, BD is their common perpendicular, ...... (Theor. III.).

Hence AB and CD are in the same plane, and the straight line BD, cutting them, makes the angles B and D two right angles.

Therefore the lines are parallel.

THEOREM VIII.

If two straight lines be parallel, and one of them be perpendicular to a plane, the other shall be perpendicular to it also.

Let AB and CD be parallel, and AB perpendicular to MN; it is required to prove that CD is also perpendicular

to MN.

Proof. Let B and D be the points of intersection of AB and CD with MN.

AB and CD are in the same plane, being parallel, and BD is in the same plane with them,

(Def. 3).

AB and CD being parallel, the angles ABD and CDB are together equal to two right angles; but since AB is perpendicular to MN, ABD is a right angle; therefore CDB is a right angle.

Because AB is perpendicular to MN, the plane ABD is also perpendicular to MN, ...... (Theor. IV.).

But it has been proved that CD is in the plane ABD, and that it is perpendicular to BD, the common section of the two planes.

Therefore CD is perpendicular to MN,

THEOREM IX.

..... (Theor. v.).

Two straight lines which are each parallel to the same

straight line are parallel to one another.

Let CD and EF be each parallel to AB; it is required to prove that they are parallel to one another.

Proof. Let the plane MN be perpendicular to AB. Then CD and EF are each perpendicular to MN, (Theor. VIII.).

......

And consequently parallel to one another, (Theor. VII.).

DEF. 8. A straight line and a plane are parallel to one another when they cannot meet, however far they may be produced.

DEF. 9. Planes which do not meet when indefinitely produced in every direction are parallel to one another. It follows from definitions 8 and 9 that if two planes are parallel to one another, any line in one of them must be parallel to the other.

THEOREM X.

If two straight lines are parallel to one another, any plane which contains one of them, but not both, is parallel to the other.