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Exercises. Planes

1. We commonly say that we live in a space of three dimensions, these dimensions being length, width, and thickness. We may, then, consider a plane as a space of how many and what dimensions? Similarly, a line is a space of how many and what dimensions?

2. Explain the meaning of the statement that a plane passes through a line; that it cuts or intersects the line. Draw a figure to illustrate each case.

3. Two lines in a plane may have no point in common, in which case they are parallel; they may have one point in common, in which case they intersect; or they may have an infinite number of points in common. Write a similar statement respecting two planes in three-dimensional space.

4. Write a statement mentioning three points in the room and describing the position of the plane determined by them. Illustrate the statement by a drawing.

5. Write a statement explaining why a three-legged stool stands firmly on the floor while a four-legged chair may not do so.

6. Write a statement describing the position of two lines in the room which are so situated that they do not determine a plane and do not meet however far produced.

7. State the geometric reason why a triangle is necessarily a plane figure while a four-sided figure in threedimensional space need not be.

8. In three-dimensional space how many different planes are determined by four points? by five points?

9. If n lines, no two of which are parallel, meet a given line l, how many planes are determined, and upon what postulate does your answer depend?

Proposition 1. Intersection of Planes

32. Theorem. If two planes meet, they intersect in a straight line.

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A

p

B

Given two planes p and q which meet.

Prove that p and q intersect in a st. line.

Proof. Since p and q meet, they must have at least one point, as A, in common. Hence they must have at least one other point, as B, in common.

AB.

§ 31, 2

Draw

Post. 1

Then

$ 3,9

because otherwise p and q would not be planes.

§ 31, 1

AB lies in both p and q,

Also, no point not on AB can be in both p and q, because p and q would then coincide instead of meeting. Hence the st. line determined by A and B contains all points common to p

and q.

.. AB is the intersection of p and q,
because this is the meaning of intersection.

Hence

p and q intersect in a st. line.

33. Perpendicular to a Plane. If a straight line which meets a plane is perpendicular to every straight line which lies in the plane and passes through the point of meeting, the line is said to be perpendicular

to the plane, and the plane is said to be perpendicular to the line.

In this figure we may have any number of planes containing 1, and in each plane we may have a perpendicular to l

α

at O. Hence we may have any number of perpendiculars, as a, b, c, to l at O.

If, as will be shown (§ 38) to be the case, a, b, c, plane m, then l is to m, and m is to l.

m

all lie in one

If we invert the above definition (§ 3, 4), we see that if a straight line is perpendicular to a plane, the line is perpendicular to every line in the plane that passes through the point of meeting.

34. Foot of a Perpendicular. The point at which a perpen-. dicular meets a plane is called the foot of the perpendicular.

35. Oblique to a Plane. If a straight line which meets a plane is not perpendicular to the plane, the line is said to be oblique to the plane, and the plane is said to be oblique to the line.

Lines which are perpendicular or oblique to a plane are called perpendiculars or obliques respectively.

When we speak of a perpendicular or an oblique from a point to a plane, we mean the line segment from the point to the plane.

36. Parallel to a Plane. If a straight line cannot meet a plane, however far each is produced, the line is said to be parallel to the plane, and the plane is said to be parallel to the line. Similarly, if one plane cannot meet another plane, however far each is produced, the planes are said to be parallel.

Proposition 2. Perpendicular to a Plane

37. Theorem. If a line is perpendicular to each of two intersecting lines at their point of intersection, it is perpendicular to the plane of the two lines.

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Given AO to OP and OR at 0, and m, the plane of OP and OR. Prove that

AO is to m.

Proof. Through O draw any other line OX in the plane m, and draw PR, cutting OP, OX, OR in P, Q, and R respectively. On AO produced take OA'= OA.

Join A and A' to P, Q, and R respectively.

Then

AP A'P and AR=A'R,

because OP and OR are each 1 to AA' at its midpoint.

.. AAPR is congruent to AA'PR.

Then

ZRPA=ZRPA'.

... APQA is congruent to APQA'.

§ 14, 2

$7,5

$ 7,2

$ 7,3

Then, since AQ = A'Q (§ 7, 2), OQ is to AA' at O. § 19, 4

Hence AO is to any line in the plane m through O, and thus is to m.

§ 33

Proposition 3. Perpendiculars to a Line

38. Theorem. Every line perpendicular to a given line at a given point lies in a plane perpendicular to the given line at the given point.

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Proof. Suppose that the plane p, determined by OA and OY, does not intersect m in OA, but intersects it in OA'.

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Hence OA lies in m, and similarly for OB, OC,... $3,9 39. Corollary. Through a given internal point there can be one and only one plane perpendicular to a given line.

Y

40. Corollary. Through a given external point there can be one and only one plane perpendicular to a given line. In a, the plane of YY' and the given point P, let PO be to YY'. In b, any other plane containing YY', let OQ be 1 to YY'. Then m, the plane of OP and OQ, is to YY' at O (§38). Now prove that m is the only plane by using Post. 7.

IP

m

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