The Elements of Plane and Solid Geometry

PARALLEL PLANES.

Proposition 7. Theorem.

512. If two straight lines are parallel, each of them is parallel to every plane passing through the other and not containing both lines.

Hyp. Let AB, CD be two || st. lines, A and MN any plane passing through

CD.

To prove Proof. The

AB || to MN.

s AB, CD lie wholly

in the plane ABCD.

... if AB meets the plane MN, it must meet it in some

pt. of the intersection CD, of the two planes.

But AB is | to CD, and so cannot meet it. ... AB cannot meet the plane MN.

... AB is || to MN.

(Hyp.)

Q.E.D.

513. COR. 1. A line parallel to the intersection of two planes is parallel to each of those planes.

514. COR. 2. Through any given straight line, a plane can be passed parallel to any other given straight line.

515. COR. 3. Through a given point a plane can be passed parallel to any two given straight lines in space.

For, draw through the given pt. 0, in the plane of the given line AB and O, the line A'B' || to AB, and in the plane of the given line CD and O, the line C'D' || to CD; then the plane of the lines A'B', C'D' is || to each of the lines AB and CD.

(512)

Proof. If the planes are not || they will meet if sufficiently produced. (489) There will then be, through a pt. of their intersection, two planes to the same st. line AB.

But this is impossible.

... MN is || to PQ.

A+

B+

(506)

Q.E.D.

517. COR. 1. If a straight line is parallel to a plane, the intersection of the plane with any plane passed through the line is parallel to the line.

Let the student show this.

518. COR. 2. If a straight line and a plane are parallel, a parallel to the line drawn through any point of the plane lies in the plane.

(517)

REM. A polygon in space may be formed by joining end to end any number of straight lines, as defined in (137). But in Plane Geometry the lines are all confined to one plane, while there is no such restriction upon polygons in space.

Proposition 9. Theorem.

519. The intersections of two parallel planes by a third plane are parallel lines.

Hyp. Let the || planes MN, PQ be cut by the plane AD into the lines AB and CD.

M B

(489)

Q.E.D.

1. Show that all the propositions in Plane Geometry which relate to triangles are true of triangles in space, however situated.

See (485).

2. Show that those propositions are not true of polygons of more than three sides situated in any way in space.

Proposition 10. Theorem.

521. A straight line perpendicular to one of two parallel planes is perpendicular to the other.

Hyp. Let MN and PQ be || planes, and let AB be to PQ.

And since AC is any line drawn through A in the plane

MN,

.. AB is to the plane MN.

(487)

Q.E.D.

522. COR. 1. Through a given point one plane can be passed parallel to a given plane, and only one.

For, if AB is to PQ, a plane passing through the pt. A, 1 to AB, is || to PQ. (516)

Also, since every plane || to PQ is to AB (521), and since only one plane can be passed through the pt. A to AB (504), therefore only one plane can be passed through a given pt. parallel to a given plane.

523. COR. 2. Two parallel planes are everywhere equally distant.

For, all st. lines to the plane PQ are also to the || plane MN (521); and being to the same plane, they are || (509); and being included between || planes, they are equal (520). Hence the planes are everywhere equally distant. (497)

524. COR. 3. If two intersecting straight lines are each parallel to a given plane, the plane of these lines is parallel to the given plane.

See (518), (516).

Proposition 11. Theorem.

525. If two angles not in the same plane have their sides respectively parallel and lying in the same direction, they are equal and their planes are parallel.

Hyp. Let s A and A' lie in the planes MN, PQ respectively, and let AB be to A'B' and AC be II to A'C'.

(1) To prove ZA = ZA'.

Proof. Take AB = A'B', and AC = A'C', and join AA', BB', CC', BC, B'C'.