POLYEDRAL ANGLES.

DEFINITIONS.

553. When three or more planes meet in a common point, they are said to form a polyedral angle at that point.

The common point in which the planes meet is the vertex of the angle, the intersections of the planes are the edges, the portions of the planes between the edges

are the faces, and the plane angles formed by the edges are the face-angles.

Thus, the point S is the vertex, the straight lines SA, SB, etc., are the edges, the planes A SAB, SBC, etc., are the faces, and the angles ASB, BSC, etc., are the face-angles of the polyedral angle S- ABCD.

S

554. The edges of a polyedral angle may be produced indefinitely; but to represent the angle clearly, the edges and faces are supposed to be cut off by a plane, as in the figure above. The intersection of the faces with this plane forms a polygon, as ABCD, which is called the base of the polyedral angle.

555. In a polyedral angle, each pair of adjacent faces forms a diedral angle, and each pair of adjacent edges forms a face-angle. There are as many edges as faces, and therefore as many diedral angles as faces.

556. The magnitude of a polyedral angle depends only upon the relative position of its faces, and is independent of their extent. Thus, by the face SAB is not meant the triangle SAB, but the indefinite plane between the edges SA, SB produced indefinitely.

557. Two polyedral angles are equal, when the face and

diedral angles of one are respectively equal to the face and diedral angles of the other, taken in the same order.

558. A polyedral angle of three faces is called a triedral angle; one of four faces is called a tetraedral angle; etc. 559. A polyedral angle is convex when its base is a convex polygon. (141)

560. A triedral angle is called isosceles when it has two of its face-angles equal; when it has all three of its faceangles equal it is called equilateral.

561. A triedral angle is called rectangular, bi-rectangular, or tri-rectangular, according as it has one, two, or three, right diedral angles.

The corner of a cube is a tri-rectangular triedral angle.

562. Two polyedral angles are symmetrical, when the face and diedral angles of one are equal to the face and diedral angles of the other, each to each, but arranged in reverse order.

Thus, the triedral angles S-ABC, S' - A'B'C' are symmetrical when the face-angles ASB, BSC, CSA are equal respectively to the face-angles A'S'B', B'S'C', C'S'A', and the diedral angles SA, SB, SC to the diedral angles S'A', S'B', S'C'.

When two polyedral angles are

AA

symmetrical, it is, in general, impossible to bring them into coincidence.

The two hands are an illustration. The right hand is symmetrical to the left hand, but cannot be made to coincide with it. The right glove will not fit the left hand, but is symmetrical to it.

563. Opposite or vertical polyedral angles are those in which the edges of one are the prolongations of the edges of the other.

Proposition 22. Theorem.

564. Two opposite polyedral angles are symmetrical.

B'

A

S

Hyp. Let S - ABC, S- A'B'C' be two opp. triedral ≤ s. To prove they are symmetrical.

Proof. Because the faces ASB and A'SB' are vertical

Also, because the diedral between two planes is the same at every pt., (528)

... the diedral s whose edges are SA, SB, etc., = respectively the diedrals whose edges are SA', SB', etc. But the edges of S- A'B'C' are arranged in the reverse order from the edges of S- ABC.

.. S-ABC is symmetrical to S- A'B'C'.

EXERCISE.

Q.E.D.

Pass two parallel planes, one through each of two straight lines which do not meet and are not parallel.

Let AB, CD be the lines: draw AE to CD, CF || to AB... plane AEB is to plane CFD.

Proposition 23. Theorem.

565. The sum of any two face-angles of a triedral angle is greater than the third.

S

Hyp. Let S-ABC be a triedral in which ASC is the greatest face .

To prove

ZASB+BSC > < ASC.

Proof. In the plane ASC draw SD,

making

ASD = ASB.

=

Draw AC cutting SD in D, take SB SD, and join AB,

But

SC is common, SB = SD, and BC > DC,

... < BSC> <DSC.

(120) (Cons.)

Q.E. D.

Adding, ASB + BSC > < ASC.

Proposition 24. Theorem.

566. The sum of the face-angles of any convex polyedral angle is less than four right angles.

Hyp. Let the convex polyedral S be cut by a plane making the section ABCDE a convex polygon.

S

C

S, and the other with their common vertex at O, and an equal number of each.

... the sum of the s of these two sets of ▲s is equal. (97)

Because the sum of any two faces of a triedral > the third,

and

.. SAE+ SAB > ZEAB,

ZSBA + SBC > < ABC, etc.

(565)

Taking the sum of these inequalities, we find that the sum of the s at the bases of the s whose common vertex is S > the sum of the s at the bases of the ▲s whose common vertex is 0.

.•. the sum of the s at S < the sum of the s at O.

... the sum of the s at S < 4 it. ≤ s.

(56) Q.E.D.