PROPOSITION XXI. THEOREM

509. Through any given straight line not perpendicular to a plane, one plane can be passed perpendicular to that given plane, and only one.

Hyp. AB is a line not to plane MN.

To prove through AB one, and only one, plane can be passed 1 to MN.

Proof. From any point C in AB, draw CD 1 to MN.
Pass a plane AE through AB and CD.

Then plane AD is to MN.

(502)

If any other plane could be passed through AB to MN, AB, their intersection, would be perpendicular to MN. (506) But this is impossible since it contradicts the hypothesis. Hence, only one plane can be drawn perpendicular to MN and passing through AB.

Q.E.D.

510. COR. The projection of a line not perpendicular to a plane upon that plane is a straight line.

Ex. 1056. If the projection of a figure upon a plane is a straight line, the figure is a plane figure.

PROPOSITION XXII. THEOREM

512. The acute angle formed by a line and its projection upon a plane is the least angle which the line makes with any line in the plane.

B

M

A

Hyp. The line AC is the projection of line AB upon plane MN, and AD is any other line drawn through A in MN.

Ex. 1058. In the diagram for Prop. XXII, if AC = 3 inches, Z BAC= 60°, and CAD = 90°, find the length of BD.

Ex. 1059. The inclinations of two parallel lines to a plane are equal.

PROPOSITION XXIII. PROBLEM

513. To draw a common perpendicular to two lines

Given. Lines AB and CD not in the same plane. Required. To draw a common perpendicular to AB and CD. Construction. Through CD draw plane MN parallel to AB. Through AB draw plane AFL to MN, intersecting MN in EF, and let G be the intersection of EF and CD.

At G, in plane AF, draw GK to EF, meeting AB in K. Then GK is the required perpendicular.

Proof. Since EF and AB are parallel, and GK is to EF, GK must be to AB.

(88) (503)

Q.E.F.

514. COR. Only one common perpendicular can be drawn to two lines not in the same plane.

For, suppose there was another common perpendicular HD,

POLYEDRAL ANGLES

515. DEF. A polyedral angle or a solid angle is formed by three or more planes meeting in a point.

516. DEF. The vertex of a polyedral angle is the common point in which the planes meet; the edges are the intersections of the planes; the faces are the planes bounded by the edges; and the face angles are the angles. formed by the edges.

Thus, if three planes ABV, BCV, and ACV meet in V, V is the vertex, VA, VB, and VC are the edges, the planes AVB, BVC, CVA are the faces, and the angles AVB, BVC, CVA are the face angles of the polyedral angle V-ABC.

V

B

The magnitude of a polyedral angle depends upon the relative positions of its faces and not upon their extent.

In a polyedral angle each pair of adjacent faces forms a diedral angle, and each pair of adjacent edges forms a face angle. The face angles and the diedral angles are called the parts of the polyedral angle.

517. DEF. A polyedral angle is convex if any section made by a plane intersecting all its faces is a convex polygon.

518. DEF. A polyedral angle is called triedral, tetraedral, etc., according as it has three, four, etc., faces.

519. DEF. A polyedral angle is called rectangular, birectangular, or tri-rectangular, according as it has one, two, or three, right diedral angles.

520. DEF. An isosceles triedral angle is one, two of whose face angles are equal.

521. Two polyedral angles are equal if the face and diedral angles of the one are respectively equal to the face and diedral angles of the other one, and all the parts are arranged in the same order (for evidently they can be made to coincide).

Two polyedral angles are symmetrical if the face and diedral angles of the one are respectively equal to the face and diedral angles of the other, and all parts are arranged in reverse order.

Thus, in Fig. 1 the triedral V-ABC = triedral ▲ V'-A'B'C' if ZAVB=ZA'V'B', Z BVC=Z B'V'C', Z CVA=Z CV'A', and diedral ▲AV= diedral A'V', diedral ▲ BV = diedral ▲ B'V', and diedral ≤ CV = diedral ≤ C'V'.

While in Fig. 2 V-ABC and V'-A'B'C' are symmetrical.

It is obvious that, in general, two symmetrical polyedral angles cannot be made to coincide.

Ex. 1060. If in diagram for 516, VA = VB=VC = AB = BC = CA, construct (by means of a plane construction) a diedral angle.

Ex. 1061. If in the same diagram ▲ AVB = ▲ BVC and AVB and AVC are given, find (by a plane construction) diedral angle VB.

Ex. 1062. Find (by a plane construction) the diedral angles of any triedral angle, if the face angles are given.