Two triedrals are either equal or symmetrical, if two diedrals and the interjacent plane face of the one are equal to two diedrals and the interjacent plane face of the other, each to each. The demonstration is similar to that in Proposition VIII., Book I. We cause thus the second triedral to coincide either with the first or with its symmetrical triedral. PROPOSITION XXXVIII. THEOREM. Two triedrals are either equal or symmetrical when the plane angles of the one are equal to the plane angles of the other, each to each. Let the angle ASB = A'S'B', BSC - B'S'C', CSA = C'S'A'. Take SL = S'L', and at the point L erect in the planes ASB, ASC, re The triangles SLM and S'L'M' are equal, having SL S'L' by construction, angle ASB A'S'B' by hypothesis, and SLM = S'L'M', being right angles. SM = = Hence S'M', and LM = L'M'. Similarly, triangle SLN = triangle S'L'N', and hence SNS'N', LN = L'N'. Therefore, the triangles MSN and M'S'N' are equal (Prop. VII., Book I.), and hence MN M'N'. Hence the two triangles LMN and L'M'N' are equal (Book I., Prop. XII.), and therefore the angle MLN = M'L'N'. But these angles measure the diedrals BSAC, B'S'A'C', respectively (Prop. XXV., Cor. 1). Hence, BSAC = B'S'A'C'. Hence (Prop. XXXVI.) the two triedrals are either equal or symmetrical. SCHOLIUM. This demonstration fails when two plane angles, ASB and ASC, are right (as will be seen from the construction). But, in that case, the edge SA is perpendicular to the plane BSC, and the edge S'A' to the plane B'S'C'. Hence, when we place B'S'C' on BSC, the perpendicular S'A' will coincide with SA (Prop. IV.), and thus the two triedrals will coincide. PROPOSITION XXXIX. THEOREM. The perpendiculars; S'C', S'A', S'B', drawn from an interior point, S', of a triedral, S, on its three faces, ASB, ASC, BSC, form a second triedral, S', the plane angles and diedrals of which are respectively the supplements of the diedrals and plane angles of the triedral S. First, we have (Prop. XXX.) angle B'S'C' 2 right angles diedral BSAC = angle C'S'A' 2 right angles - diedral CSBA Moreover, the three planes, B'S'C', C'S'A', A'S'B' are perpendicular, respectively, to the edges, SA, SB, SC (Prop. XXX.); hence, conversely, the triedral S has the same properties with regard to the triedral S' that S' has with regard to S. Hence, we have angle ASB = 2 right angles diedral A'S'C'B' A angle CSA 2 right angles - diedral C'S'B'A'. SCHOLIUM 1. These triedrals, S and S', are said to be supplementary to each other, or the two taken together are called supplementary triedrals. SCHOLIUM 2. If perpendiculars be drawn, at S', to the planes ASB, ASC, BSC, so that they shall fall on the same sides of these planes with the edges SC, SB, SA, respectively, opposite to these planes, the triedral thus formed will be also supplementary to S'-ABC, since it will fulfil all the conditions of S'-A'B'C'. PROPOSITION XL. THEOREM. Two triedrals are either equal or symmetrical when the three diedrals of the one are equal to the three diedrals of the other, each to each. If we construct the triedrals supplementary to these two, these new triedrals will (Prop. XXXIX.) have the plane angles of the one equal to the plane angles of the other, and hence (Prop. XXXVIII.) the diedrals of the one will be equal to the diedrals of the other, each to each. Therefore (Prop. XXXIX.), the given triedrals will have their plane angles equal, each to each. And hence, since all the parts of the one are equal to all the parts of the other, each to each, these two triedrals will either be equal or symmetrical. PROPOSITION XLI. THEOREM. The sum of the three diedrals of a triedral is always greater than two right angles and less than six. Since each diedral of a triedral is equal to two right angles, minus the opposite plane angle of the supplementary triedral, the sum of the three diedrals is equal to six right angles, minus the sum of the plane angles of the supplementary triedral. XXXIII.) is less than four right angles. above is greater than two right angles. But this sum (Prop. Therefore, the difference The second part is evident, since each diedral is less than two right angles. EXERCISES ON BOOK V. THEOREMS. 1. If we draw a plane perpendicular to a straight line at its middle point, Ist. Any point on this plane will be equally distant from the extremities of the line. 2d. Any point not on this plane will be unequally distant from these extremities. 2. If any angle, BAC, revolve about its side AB, returning to its first position, every point of AC will describe a circumference, the plane of which is perpendicular to AB. 3. Two planes which are parallel to the same straight line, are either parallel to each other or their intersection is parallel to this line. 4. Two planes which are parallel to two planes which intersect each other, will also intersect, and the line of intersection of the first planes will be parallel to the line of intersection of the second. 5. If from the projections, P and Q, of the same point, O, upon two planes which intersect, perpendiculars be drawn to the line of intersection, they will meet it at the same point. 6. Conversely, if the perpendiculars from two points, P and Q, on the common intersection of the two planes in which they lie, meet that intersection at the same point, then P and Q are the projections of the same point, O, in space on these two planes. 7. Two planes perpendicular to the same plane, P, and containing two lines, AB, A'B', parallel to each other, are parallel. Show also that this is not true if the lines AB and A'B' are perpendicular to the plane P. 8. The projections of two parallel lines on the same plane are parallel. Show also that the converse of this is not true. 9. The angles which two parallel lines oblique to a plane make with that plane, are equal. 10. If two planes, P and Q, intersect each other, a straight line in P, perpendicular to their common intersection, makes a greater angle with the plane Q than any other straight line drawn in P. NOTE. This line through any point in P, perpendicular to the common intersection, is called the line of greatest inclination to the plane Q. 11. A straight line and plane which are perpendicular to the same plane are parallel. 12. Any point on the plane bisector of a diedral angle is equally distant from the faces of the diedral, and any point within the diedral not on this plane bisector is unequally distant from the faces of the diedral. 13. The perpendiculars let fall from the same point on planes which have a common intersection are all in the same plane. Show also that this is true of perpendiculars let fall from the same point on planes whose intersections are parallel. 14. If, through one of the diagonals of a parallelogram, we pass any plane, the perpendiculars let fall from the extremities of the other diagonal on this plane will be equal. 15. In any triedral, the three planes perpendicular to the three faces, and containing the bisectrices of the plane angles, meet in the same straight line. 16. The three planes which bisect the diedral angles of a triedral meet in the same straight line. 17. The three planes containing the three edges of a triedral and the bisectrices of the opposite plane angles meet in the same line. 18. The three planes drawn through the three edges of a triedral, perpendicular to the opposite faces, meet in the same line. 19. Show that if, in the place of the triedral, we have three planes whose intersections two and two are parallel, the last four theorems are equally true. 20. If through the middle point of the perpendicular to two straight lines, not situated in the same plane, we draw a plane parallel to these two lines, this plane will bisect every line which joins the two lines. 21. When a straight line is parallel to a plane, the shortest distance from this line to any line of the plane not parallel to the first line is constant. 22. If three straight lines, A, B, C, not situated in the same plane, lie two and two in the same planes, these lines either all meet in the same point or are all parallel. 23. Every section of a rectangular triedral, by a plane perpendicular to one of its edges, is a right angled triangle. 24. In any triedral the greater plane face lies opposite the greater diedral, and conversely. |