GEOMETRY.

BOOK V.

OF PLANES AND THEIR ANGLES.

DEFINITIONS.

1. A straight line is perpendicular to a plane, when it is per pendicular to every straight line of the plane which it meets. The point at which the perpendicular meets the plane, is called the foot of the perpendicular.

2. If a straight line is perpendicular to a plane, the plane is also said to be perpendicular to the line.

3. A line is parallel to a plane when it will not meet that plane, to whatever distance both may be produced. Conversely, the plane is then parallel to the line.

4. Two planes are parallel to each other, when they will not meet, to whatever distance both are produced.

5. If two planes are not parallel, they intersect each other in a line that is common to both planes: such line is called their common intersection.

6. The space included between two planes is called a diedral angle: the planes are the faces of the angle, and their intersection the edge. A diedral angle is measured by two lines, one in each plane, and both perpendicular to the common intersection at the same point.

This angle may be acute, obtuse, or a right angle. When it is a right angle, the planes are said to be perpendicular to each other.

then will the angle DCE measure the inclination between the two planes.

It should be remembered that the line EC is directly over the line CD.

7. A polyedral angle is the angular space included between several planes meeting at the same point.

Thus, the polyedral angle S is formed by the meeting of the planes ASB, BSC, CSD, DSA.

8. The angle formed by three planes is called a triedral angle.

THEOREM I.

D

B

Two straight lines which intersect each other, lie in the same plane, and determine its position.

Let AB and AC be two straight lines which intersect each other at A.

Through AB conceive a plane to be passed, and let this plane be turned around AB until it embraces the point C: the plane will then contain the two

B

C

lines AB, AC, and if it be turned either way it will depart from the point C, and consequently from the line AC. Hence,

Of Planes.

the position of the plane is determined by the single condition of containing the twe straight lines AB, AC.

Cor. 1. A triangle ABC, or three points A, B, C, not in a straight line, determine the position of a plane.

Cor. 2. Hence, also, two parallels AB, CD determine the position of a plane. For drawing EF, we see that the plane of the two straight lines AE, EF is that of the parallels AB, CD.

THEOREM II.

A perpendicular is the shortest line which can be drawn from point to a plane.

Let A be a point above the plane DE, and AB a line drawn perpendicular to the plane: then will AB be shorter than any oblique line AC.

For, through B, the foot of the perpendicular, draw BC to the point where the oblique line AC meets the plane.

Now, since AB is perpendicular to the plane, the. angle ABC will be a

A

B

E

right angle (Def. 1.), and consequently less than the angle C: therefore, AB, opposite the angle C, will be less than AC. opposite the angle B (Bk. I. Th. xi).

Of Planes.

Cor. It is evident that if several lines be drawn from the point A to the plane, that those which are nearest the perpen dicular AB, will be less than those more remote.

Sch. The distance from a point to a plane is measured on the perpendicular: hence, when the distance only is named, the shortest distance is always understood.

THEOREM III.

The common intersection of two planes is a straight hne.

Let the two planes AB, CD, cut each other. Join any two points E

and F, in the common intersection, A by the straight line EF. This line will lie wholly in the plane AB, and also wholly in the plane CD (Bk. I. Def. 7); therefore, it will be in both planes at once, and consequently, is their common intersection.

E

B

THEOREM IV.

A straight line which is perpendicular to two straight lines u their point of intersection, will be perpendicular to the plane of those lines.

Let the line PA be perpendicular to the two lines AD, AB: then will it be perpendicular to the plane BC which contains them.

For, if AP is not perpendicular to the plane BC, suppose a plane

D

B

Problems.

PROBLEM III.

To find a fourth proportional to the lines A, B, and C. Place two of the lines forming an angle with each other at A; that is, Bmake AB equal to A, and AC equal

B; also, lay off AD equal to C.

A

E

Then join BC, and through D draw

DE parallel to BC, and AE will be the fourth proportional sought.

For, since DE is parallel to BC, we have

AB : AC :: AD : AE;

therefore, AE is the fourth proportional sought.

PROBLEM IV.

To find a mean proportional between two given lines, A and B.

AC, and it will be the mean proportional sought (Th. xviii. Cor).

PROBLEM V.

To make a square which shall be equivalent to the sum of two

BC equal to B: then draw AC and the square described on AC will be equivalent to the squares on A and B (Th. xii).