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SOLID GEOMETRY

BOOK VI

LINES AND PLANES IN SPACE - POLYEDRAL

ANGLES

Geometry of space or solid geometry treats of figures whose elements are not all in the same plane.

(15) 453. Def. A plane is a surface such that a straight line joining any two points in it lies entirely in the surface. (11)

A plane is determined by given points or lines, if only one plane can be drawn through these points or lines.

PROPOSITION I. THEOREM 454. A plane is determined : (1) By a straight line and a point without the line. (2) By three points not in the same straight line. (3) By two intersecting straight lines. (4) By two parallel straight lines.

[graphic]

(1) To prove that a plane is determined by a given straight line AB and a given point C.

Turn any plane EF passing through AB about AB as an axis until it contains C.

If the plane, so obtained, be turned in either direction, it would no longer contain C.

Hence, the plane is determined by AB and C.

(2) To prove that three given points A, B, and C determine a plane.

Draw AB. Then AB and C determine the plane. (Case 1.)

B

(3) To prove that two intersecting straight lines AB and AC determine a plane.

[Proof by the student.]

A

(4) To prove that two parallel lines AB and CD determine a plane.

The parallel lines AB and CD lie in the same plane by definition.

Since AB and the point C determine a plane, the two parallels determine a plane.

455. Cor. The intersection of two planes is a straight line.

For the intersection cannot contain three points not in a straight line, since only one plane can be passed through three such points.

456. DEF. The foot of the line intersecting a plane is the point of intersection.

457. DEF. A straight line is perpendicular to a plane if it is perpendicular to every line drawn through its foot in the plane.

458. DEF. A plane is perpendicular to a line if the line is perpendicular to the plane.

459. DEF. A straight line and a plane are parallel if they do not meet, however far they may be produced.

460. DEF. Two planes are parallel if they do not meet, however far they may be produced.

PROPOSITION II. THEOREM 461. If a straight line is perpendicular to each of two lines at their point of intersection, it is perpendicular to the plane of those lines.

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Hyp. AB is I to BC and BD at B.
To prove AB is I to the plane MN containing BC and BD.
Proof. In plane MN draw any line BE through B.

Draw CD meeting BE in E and produce AB to F, so that BF = AB.

Draw AC, AE, AD, CF, EF, and DF.

Since BC is I to AF at its midpoint, AC= FC.
Similarly

AD= DF.

CD is common.
Hence
A ACD = A CDF.

(Why ?) .. Z ACD= ZFCD, and A ACE = AFCE.

(Why?) .. EA= EF. Then, since E and B are respectively equidistant from A and F, BE is I to AF.

(Why ?) Hence, AB is I to any line passing through its foot in the plane MN, and is therefore I to the plane.

Q.E.D.

PROPOSITION III. THEOREM 462. All the perpendiculars that can be drawn to a straight line at a given point lie in a plane perpendicular to the line at that point.

Hyp. AB is I to BC, BD, and BE at B.
To prove BC, BD, and BE lie in a plane I to AB.
Proof. Pass a plane MN through BD and BE.
Then
AB is I to MN.

(Why ?)
Let the plane containing AB and BC intersect MN in BC".
Then
BC" is I to AB.

(Why ?) But

BC is I to AB. But in the plane ABC, only one perpendicular can be drawn upon AB at B. Hence BC and BC" coincide, and BC lies in the plane MN.

Q.E.D. 463. CoR. 1. Through a point C, to pass a plane perpendicular to a line AB, draw CD perpendicular to AB in plane ABC, meeting AB in D. At D, in any other plane than ABC, draw DF perpendicular to AB. Then the two perpendiculars DC and DF determine a plane perpendicular to AB.

464. Cor. 2. At a given point in a straight line, one and only one plane can be drawn perpendicular to the straight line.

465. Cor. 3. Through a point without a straight line, one plane and only one can be drawn perpendicular to that line.

Ex. 1033. The locus of a point in space equidistant from the ends of a line is a plane perpendicular to the line at its midpoint.

PROPOSITION IV. THEOREM 466. If three planes intersect each other, the lines of intersection either meet in a point or are parallel.

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Hyp. AD, CF, and BE are three planes intersecting respectively in CD, AB, and EF.

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