GEOMETRY OF PLANES.*

DEFINITIONS.

1. THE angle formed by two lines not in the same plane is the angle formed by one of them with a line drawn through any point of it parallel to the other.

2. A plane is a surface in which, if any two points be taken, the straight line which joins these points will be wholly in that surface.

3. A straight line is said to be perpendicular to a plane when it is perpendicular to all the straight lines in the plane which pass through the point in which it meets the plane.

This point is called the foot of the perpendicular.

4. The inclination of a straight line to a plane is the acute angle contained by the straight line, and another straight line drawn from the point in which the first meets the plane, to the point in which a perpendicular to the plane, drawn from any point in the first line, meets the plane.

5. A straight line is said to be parallel to a plane when it can not meet the plane, to whatever distance both be produced.

6. It will be proved in Prop. 2, that the common intersection of two planes is a straight line; this being premised,

The angle contained by two planes, which cut one another, is measured by the angle contained by two straight lines drawn, one in each of the planes, perpendicular to their common intersection at the same point.

This angle may be acute, right, or obtuse.

* Students intending to pursue that subject, may here with advantage take up Plane Trigonometry before going on with the Geometry of Planes and Solids.

If it be a right angle, the planes are said to be perpendicular to each other.

The angle formed by two planes is called diedral. 7. Two planes are parallel to each other when they can not meet, to whatever distance both be produced.

8. A plane is ordinarily represented, in a diagram, by a parallelogram, and called by the two letters at the opposite (diagonally) angle. This plane, which must be conceived to be indefinitely extended, divides space into two indefinite portions called regions.

Two or more planes are represented in their relative position not accurately, but by a sort of perspective.

PROP. I.

A straight line can not be partly in a plane and partly out of it.

For, by def. (1), when a straight line has two points common with a plane, it lies wholly in that plane.

PROP. II.

If two planes cut each other, their common intersection is a straight line.

Let the two planes AB, CD cut each

other, and let P, Q be two points in A P their common section.

Join P, Q;

Then, since the points P, Q are in the same plane AB, the straight line PQ which joins them must lie wholly in that plane (def. 2).

For a similar reason, PQ must lie C wholly in the plane CD.

B

.. The straight line PQ is common to the two

planes, and is .. their common intersection.

Note. In this and the following diagrams concealed lines are drawn dotted.

PROP. III.

Any number of planes may be drawn through the same straight line.

For let a plane, drawn through a straight line, be conceived to revolve round the straight line as an axis. Then the different positions assumed by the revolving plane will be those of different planes drawn through the straight line

PROP. IV.

One plane, and one plane only, can be drawn,

1o. Through a straight line, and a point not situated in the given line.

2°. Through three points which are not in the same straight line.

3°. Through two straight lines which intersect each other.

4°. Through two parallel straight lines.

1. For if a plane be drawn through the given line, and be conceived to revolve round it as an axis, it must in the course of a complete revolution pass through the given point, and so assume the position enounced in 1o.

Also, one plane only can answer these conditions, for if we suppose a second plane passing through the same straight line and point, it must have at least two intersections with the first, which is evidently impossible.

2. Join two of the points; this case is then reduced to the last.

3. Take a point in each of the lines which is not the point of intersection; join these two points; the case is now the same as the two former.

4. Parallel straight lines are in the same plane, and, by the first case, one plane only can be drawn through either of them, and a point assumed in the other.

Cor. Hence the position of a plane is determined by, 1. A straight line, and a point not in the given straight line.

2. A triangle, or three points not in the same straight line.

3. Two straight lines which intersect each other. 4. Two parallel straight lines.

PROP. V.

If a straight line be perpendicular to two other straight lines which intersect at its foot in a plane, it will be perpendicular to every other straight line drawn through its foot in the same plane, and will therefore be perpendicular to the plane.

Let XZ be a plane, and let the straight line PQ be perpendicular to the two straight lines AB, CD which intersect in Q in the plane XZ.

We shall prove that PQ will be perpendicular to any other straight line EF, drawn through Q in the plane XZ.

X

D

E

B

C

P

K

Q

F

H

Draw through any point K in QE a straight line GH, such that GK KH. (See exercise 4, p. 72.) Join P, G; P, K; P, H;

=

Then, since GH, the base of the ▲ GQH, is bisected in K;

.. (by th. 30), GQ' + HQ = 2GK' + 2QK'... (1) Similarly, since GH, the base of A GPH, is bisected in K;

=

2GK'+2PK'.

.. GP2+HP2 But the triangles PQG, PQH are right-angled at Q; the last expression becomes

PQ+ GQ + PQ2 + HQ2 = 2GK' + 2PK'... (2) Taking (1) from (2), there remains

2PQ'=2PK'— 2QK',

.. dividing by 2, and transposing,

PQ'+QK'= PK'.

Hence the triangle PQK is right-angled at Q, for in the right-angled triangle alone the sum of the squares of two of the sides is equal to the square of the third (see theorems 26, 28, 29.)

In like manner, it may be proved that PQ is at right angles to every other straight line passing through Q in the plane XZ.

PROP. VI.

A perpendicular is the shortest line which can be drawn to a plane from a point without.

Let PQ be perpendicular to the

plane XZ;

From P draw any other straight line PK to the plane XZ;

Then PQ

PK.

In the plane XZ draw the straight line QK, joining the points Q, K. Then, since the line PQ is perpendicular to the plane XZ, it is per

P

X

pendicular to QK, a line of the plane; and .:. PQ is less than PK. (Geom. Theor., 17.)

PROP. VII.

P

Oblique lines equally distant from the perpendicular are equal, and, if two oblique lines be unequally distant from the perpendicular, the more distant is the larger. That is, if QG, QH, QK . . . . are all equal, then PG, PH, PK . . . ... are all equal; and if QI be greater than QG, then PI is greater than PG. For the three right-angled triangles PQG, PQH, PQK having two sides in each equal, the third sides are equal (th. 26, corol. 2); and since PH < PI (th. 17),.. PG < PI.*

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I

This Proposition affords a method of finding the foot of a perpendicular to a givea plane from a point without. With a straight F