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PROPOSITION XXII. THEOREM. 479. If two straight lines be not in the same plane, one and only one common perpendicular to the lines can be drawn.
Let A B and C D be two given straight lines not in
the same plane. We are to prove one and only one common perpendicular to ihe two lines can be drawn.
Since A B and C D are not in the same plane they are not II,
§ 474 (two Ils lie in the same plane). Through the line A B pass the plane M N || to C D.
Since C D is ll to the plane M N, all its points are equally distant from the plane M N;
§ 432 hence C'D', the projection of the line C D on the plane MN, will be Il to CD,
§ 76 and will intersect the line A B at some point as C".
Now since CC' is the line which projects the point C upon the plane M N, it is I to the plane MN;
§ 435 hence CC' is I to C'D' and A B,
§ 430 (if a line be I to a plane, it is I to every line drawn through its foot in the
$ 67 ..CC' is the common I to the lines C D and A B.
Moreover, line CC' is the only common I. For, if another line E B, drawn between A B and C D, could be I to A B and C D, it would also be I to a line B G drawn Il to C D in the plane MN,
$ 67 and hence I to the plane MN.
§ 449 But EH, drawn in the plane C D' || to C C', is I to the plane M N.
§ 457 Hence we should have two ls from the point E to the plane MN, which is impossible,
$ 447 .:: CC' is the only common I to the lines C D and A B.
Q. E. D.
ON POLYHEDRAL ANGLES.
480. DEF. A Polyhedral angle is the extent of opening of three or more planes meeting in a common point.
Thus the figure S-A B C DE, formed by the planes A SB, BSC, CSD, DS E, ESA, meeting in the common point S, is a polyhedral angle.
The point S is the vertex of the angle.
The intersections of the planes AFSA, S B, etc., are its edges.
The portions of the planes В bounded by the edges are its faces.
The plane angles A SB, BSC, etc., formed by the edges are its face angles.
481. DEF. Polyhedral angles are classified as trihedral, quadrahedral, etc., according to the number of the faces.
482. DEF. Trihedral angles are rectangular, bi-rectangular, or tri-rectangular, according as they have one, two, or three right dihedral angles.
483. Def. Trihedral angles are scalene, isosceles, or equilateral, according as the face angles are all unequal, two equal, or three equal.
484. Def. A polyhedral angle is convex, if the polygon formed by the intersections of a plane with all its faces be a convex polygon.
485. DEF. Two polyhedral angles are equal when they can be applied to each other so as to coincide in all their parts.
Since two equal polyhedral angles coincide however far their edges and faces be produced, the magnitude of a polyhedral angle does not depend upon the extent of its faces. But, in order to represent the angle in a diagram, it is usual to pass a plane, as
A B C D E, cutting all its faces in the straight lines, A B, BC, etc.; and by the face A S B is meant the indefinite surface included between the lines S A and S B indefinitely produced.
486. Def. Two polyhedral angles are symmetrical if they have the same number of faces, and the successive dihedral and face angles respectively equal but arranged in reverse order. Thus, if the edges AS, BS,
СІ ВІ etc., of the polyhedral angle, S-A B C D, be produced, there is
DA fornied another polyhedral angle, S-A' B' C'D', which is symmetrical with the first, the vertex S being the centre of symmetry.
If we take S A = SA, and through the points A and A' the Af parallel planes A B C D and )
C A' B' C' D be passed, we shall
7 be passed we shoulB have S B = SB, SC" = SC, etc. For if we conceive a third parallel plane to pass through S, then A A, B B', etc., are divided proportionally, § 469. And if any one of them be bisected at S, the others are also bisected at S. Hence, the points A', B', etc., are symmetrical with A, B, etc.
Moreover, the two symmetrical polyhedral angles are equal in all their parts. For their face angles A SB and A' S B', BSC and B'SC" are equal each to each, being vertical plane angles. And the dihedral angles formed at the edges SA and SA', SB and S B', are equal each to each, being vertical dihedral angles.
Now if the polyhedral angle S-A' B'C'D be revolved about the vertex S until the polygon A' B' C D is brought into the position abcd, in the same plane with A B C D, it will be evident that while the parts A SB, BSC, etc., succeed each other in the order from left to right, the corresponding equal parts a Sb, Sc, etc., succeed each other in the order from right to left. Hence the two figures cannot be made to coincide by superposition, but are said to be equal by symmetry.
PROPOSITION XXIII. THEOREM. 487. The sum of any two face angles of a trihedral angle is greater than the third. s
Let S-ABC be a trihedral angle in which the face
angle ASC is greater than either angle A SB or
Through any point D of SD draw any straight line A DC cutting A S and SC.
Take SB= S D.
Cons. ZASD=LAS B.
Cons. .:. A ASD=A ASB,
Subtract the equals A B and A D.
BC > DC.
BC > DC,
Q. E. D.
PROPOSITION XXIV. THEOREM. 488. The sum of the face angles of any convex polyhedral angle is less than four right angles.
Let the polyhedral angle S be cut by a plane, mak
ing the section A B C D E a convex polygon.
We are to prove 2 ASB + ZBS C etc. < 4 rt. 4.
From any point 0 within the polygon draw 0 A, 0 B, OC, OD, O E.
The number of the A having their common vertex at o will be the same as the number having their common vertex at S.
... the sum of all the Is of the A having the common vertex at S is equal to the sum of all the 6 of the A having the common vertex at 0. But in the trihedral Es formed at A, B, C, etc.
ZSA E + L SAB>20A E + LO AB, § 487 (the sum of any two face & of a trihedral Z is greater than the third). and ZSBA + ZSBC > LOBA + 2OBC. $ 487
.. the sum of the < at the bases of the A whose common vertex is S is greater than the sum of the Is at the bases of the A whose common vertex is 0.
... the sum of the As at S is less than the sum of the < at 0. But the sum of the 6 at 0 = 4 rt. 4.
§ 34 ... the sum of the < at S is less than 4 rt. k.
Q. E, D,