Write the proposition, and call it Prop. XXI.

436.

Cor. With what line in the plane M N does the line A B make the greatest angle?

EXERCISE.

429. The projections of three lines on the same plane are parallel and of equal length. Can you draw any definite conclusions concerning the three lines?

Can you

437.

PROPOSITION XXII.

illustrate from the room that two straight lines in different planes may have a perpendicular between them? Suppose A B and C D two straight lines not in the same plane.

(1) Can we find a perpendicular between the lines? (2) How many perpendiculars are there?

Sug. Let PQ be a plane passed through C D and || to A B. Project A B on P Q. How does E F compare with A B? Is E F to CD? Why? Call their intersection G Does A B and its projection determine a plane? At G erect LG H in the projecting plane.

Pupil complete proof. Can you prove (2)? [§ 57.]

438.

Cor. What is the shortest distance between two lines not in the same plane?

439.

PROPOSITION XXIII.

Given: The diedral s J A B D, J' A'B' D' and their planes CA J and C' A' J'.

To show that the diedral s are to each other as their plane Zs.

and let

of

(1) Suppose JAC and J'A' C' are commensurable GA J be the common unit. Express the relation CAJ and C'A' J'.

Can you pass planes through A G and A B? A F and A B? A'F' and A' B'? How are the diedral s formed related? Can you now show that the diedrals are to each other as their planes when the planes are commensurable? (2) Suppose the unit plane G A J is contained twice with a remainder in J" A" B" K. If we let the unit continually decrease, what does / J" A" C"? Considering the

diedrals formed from these planes and decreasing the indefinitely, what does J" A" B" C" ?

unit diedral

POLYEDRAL ANGLES.

DEFINITIONS.

440.

When three or more planes meet in a point, they form a polyedral angle or polyedral.

[Thus the planes V A B, V BC, V CD, V D A, meeting in the point V, form a polyedral angle. The point at which the planes mee

meet

A

3

C

is called the vertex of the polyedral; the intersection of the planes are the edges; the planes are called the faces; and the angles A V B, B V C, etc., are called the face angles of the polyedral.]

Note. As in other problems, the planes are of indefinite extent, but to show the relation of the edges in a figure it is clearer to have a plane cutting the edges.

441.

A polyedral angle bounded by three faces is called a triedral angle; if bounded by four faces, it is called a tetraedral angle.

442.

In the figure in § 440, A B C D is called a section. How is it formed? If the section is a convex polygon, the polyedral is a convex polyedral. What is a concave polyedral?

443.

The parts of any polyedral angle are its face angles and its diedral angles. Illustrate with a pyramid.

444.

The magnitude of a polyedral angle depends entirely upon the divergence of its faces.

445.

Two polyedral angles which have the face angles and diedral angles of one respectively equal to the homologous face angles and diedral angles of the other and arranged in the same order are said to be equal.

ss

C

The face angles A V C, C V B, B V A are equal, respectively, to A' V' C', C' V' B', B' V' A', and the diedral angles VA, VC, V B, to V' A', V' C', V' B'; hence we may apply one to the other and they will coincide in all their parts; hence the polyedrals are equal.

446.

Polyedral angles which have their face angles and their diedral angles equal, each to each, and arranged in reverse order, are said to be symmetrical.

A

་

The triedral angles V - A B C and V' – A' B' C' are symmetrical if the face angles A V B, B V C, CV A are equal, respectively, to the face angles A' V' B', B' V' C', C' V' A', and the diedral angles V A, V B, V C equal, respectively, the diedral angles V' A', V' B', V' C.

Observe the position of the faces.

[Note.-The two hands or the two feet or the two sides of the face illustrate symmetrical solids. Are the right shoe and the left shoe equal? What should be said about the right glove and the left glove?]

447.

Two polyedrals are vertical when the edges of one are the prolongations of the edges of the other.

448.

PROPOSITION XXIV.

Can you prove vertical polyedrals symmetrical? [Hint.-What are the parts of a polyedral? Are the corresponding vertical parts equal? What is the order of the equal parts?]

Write the proposition, and call it Prop. XXIV.