Proposition 7. Perpendiculars to a Plane

47. Theorem. Two lines perpendicular to the same plane are parallel.

Proof. Draw AD and BD, and in m draw PQL to BD at D, making DP=DQ. Draw AP, AQ, BP, BQ.

By congruent ▲ (§ 7, 5) prove that ADP=≤ ADQ=90°, and then that BD, CD, and AD lie in the same plane (§ 38). Then prove that AB also lies in this plane (§ 3, 9), and then that AB and CD are each to BD (§ 33).

.. AB is I to CD.

$ 9,9

48. Corollary. If one of two parallel lines is perpendicular to a plane, the other is also perpendicular

to the plane.

For if through any point O of b a line is drawn to m, how is it related to a (§ 47)? Now apply § 9, 2.

49. Corollary. If two lines are parallel to a third line, they are parallel to each other.

For if b is to m, so are a and c (§ 48).

Proposition 8. Parallel Lines

50. Theorem. If two lines are parallel, every plane containing one and only one of the lines is parallel to the other.

Given the lines AB and CD, and the plane m containing ' CD but not AB.

Prove that

m is to AB.

Proof.

AB and CD determine a plane n,

§ 31, 1

and

Hence,

AB lies in n, however far each is produced.
if AB meets m, it meets CD.

$ 3,9

$ 32

Since

that is,

m is to AB.

AB cannot meet CD,

AB cannot meet m;

51. Corollary. Through either of two lines not in the

$ 9, 1

$ 36

plane m, which is determined by CD and CX, with respect to the line AB? Why can there be only one such plane? Lines placed like AB and CD in this figure are called skew lines.

52. Corollary. Through a given point one and only one plane can pass parallel to each of two given lines not in the same plane.

Let P be the given point and AB and CD the given lines. If, now, we construct through P the line A'B' || to AB, and the line C'D' || to CD, these lines determine the plane m. Then prove that m is to AB and CD, and that no other such plane is possible through P.

In the above figure, the lines AB and CD are said to form an angle, although they do not meet. This angle is defined

A

B

m

D

B'

as the C'PB', but the concept is rarely used in elementary geometry.

Exercises. Lines and Planes

1. State the geometric principle by which we know that a straight edge results from folding a piece of paper.

2. In a given plane what is the locus of points equidistant from two parallel lines in the plane? Given two parallel planes instead of two parallel lines, what is the corresponding locus in a space of three dimensions? Draw the figures but give no proofs.

3. If a given line is parallel to a given plane, the intersection of the plane with any plane passed through the given line is parallel to that line.

4. If a given line is parallel to a given plane, a line parallel to the given line drawn through any point of the plane lies in the plane.

5. If equal oblique lines are drawn from a given external point to a plane, they make equal angles with lines drawn from the points where the oblique lines meet the plane to the foot of the perpendicular drawn from the given point to the plane.

Proposition 9. Parallel Planes

53. Theorem. Two planes perpendicular to the same line are parallel.

m

ӨӨ

Given the planes m and n, each to the line 1.

Proof. If m is not I to n, it must meet n, in which case we should have two planes through a point in their intersection both to l.

Since this is impossible (§ 40), m is I to n.

The following corollary, which is analogous to the Postulate of Parallels (§ 9, 2), may be assumed, if desired, without proof.

54. Corollary. Through a given external point one and only one plane can pass parallel to a given plane.

If P is the point and m is the plane, as shown in the left-hand figure, there is only one line PQ that is 1 to m (§ 42). Through P there is one and only one plane

tain some line PP' that would meet its projection QQ ́ in m. Then n would not be II to m (§ 36). Hence only one plane through P is || to m.

Exercises. Review

1. If from the foot of a perpendicular to a plane a line is constructed at right angles to any line in the plane, the line drawn from its intersection with the line in the plane to any point on the perpendicular is perpendicular to the line in the plane.

2. If two perpendiculars extend from a given external point to a plane and to a line in that plane respectively, the line joining the feet of the two perpendiculars is perpendicular to the given line.

3. From two vertices of a triangle perpendiculars are constructed upon the opposite sides. From the intersection of these perpendiculars there is a perpendicular to the plane of the triangle. Prove that a line drawn to any vertex of the triangle from any point on this perpendicular is perpendicular to the line drawn through that vertex parallel to the opposite side.

4. Find the point in a plane to which lines may be drawn from two given external points on the same side of the plane so that their sum shall be the least possible.

From one point A suppose that a line AO is 1 to the plane and that it is produced to A', making OA'= OA. Connect A' and the other point B by a line cutting the plane at P. Then AP+PB is the least sum.

5. If three equal oblique lines are drawn from an external point to a plane, the perpendicular from the point to the plane meets the plane at the center of the circle circumscribed about the triangle which has for its vertices the points where the oblique lines meet the plane.

6. State and prove the propositions of plane geometry corresponding to §§ 47, 48, and 49. Why do not the proofs of those propositions apply to the corresponding propositions of solid geometry?