triedral

the number of triangles, in each set, is the same, it follows that these sums are equal. But in the angle whose vertex is B, we have (P. XIX.),

ABS + SBC > ABC;

and the like may be shown at each of the other vertices, C, D, E, A: hence, the sum of the angles at the bases, in the triangles whose common vertex is S, is greater than the sum of the angles at the bases, in the set whose common vertex is 0: therefore, the sum

of the vertical angles about S, is less than the sum of the angles about 0: that is, less than four right angles; which was to be proved.

Scholium. The above demonstration is made on the supposition that the polyedral angle is convex, that is, that the diedral angles of the consecutive faces are each less than two right angles.

If the plane angles formed by the edges of two triedral angles are equal, each to each, the planes of the equal angles are equally inclined to each other.

Let S and T be the vertices of two triedral angles, and let the angle ASC be equal to DTF, ASB to DTE, and BSC to ETF: then the planes of the equal angles are equally inclined to each other.

For, take any point of SB, as B, and from it draw in the two faces ASB and CSB, the lines BA and BC, respectively perpendicular to SB: then the angle ABC measures the inclination of these faces. Lay off TE equal to SB,

and from E draw in the faces DTE and FTE, the lines ED and EF, respectively perpendicular to TE: then the angle DEF measures the inclination

In like manner, it may be shown that BC is equal to EF, and CS to FT. The triangles ASC and DTF, have the angle ASC equal to DTF, by hypothesis, the side AS equal to DT, and the side CS to FT, from what has just been shown; hence, the triangles are equal in all respects, and consequently, AC is equal to DF. Now, the triangles ABC and DEF have their sides equal, each to each, and consequently, the corresponding angles are also equal; that is, the angle ABC is equal to DEF: hence, the inclination of the planes ASB and CSB, is equal to the inclination of the planes DTE and FTE. In like manner, it may be shown that the planes of the other equal angles are equally inclined; which was to be proved.

Cor. If the plane angles ASB and BSC are equal, respectively, to the plane angles DTE and ETF, and the inclination of the faces ASB and BSC is equal to that of the faces DTE and ETF, then are the remaining plane angles, ASC and DTF, equal to each other.

Scholium 1. If the planes of the equal plane angles are like placed, the triedral angles are equal in all respects, for they may be placed so as to coincide. If the planes of the equal angles are not similarly placed, the triedral angles are said to be angles equal by symmetry, or symmetrical

triedral angles. In this case, they may be placed so that two of the homologous faces shall coincide, the triedral angles lying on opposite sides of the plane, which is then called a plane of symmetry. In this position, for every point on one side of the plane of symmetry, there is a corresponding point on the other side.

Scholium 2. If the plane angles ASB and DTE are equal to each other, and the inclination of the face ASB to each of the faces BSC and ASC is equal, respectively, to the inclination of DTE to each of the faces ETF and DTF, then are the plane angles BSC and CSA equal, respectively, to the plane angles ETF and FTD. For, place the plane angle ASB upon its equal DTE, so that the point S shall coincide with T, the edge SA with TD, and the edge SB with TE, then will the face BSC take the direction of the face ETF, and the edge SC will lie somewhere in the plane ETF; the face ASC will take the direction of the face DTF, and the edge SC will lie somewhere in the plane DTF. Since SC is at the same time in both the planes ETF and DTF, it must be on their intersection (P.. III.) hence, the plane angles BSC and CSA coincide with and are equal, respectively, to ETF and FTD.

If the triedral angle whose vertex is S can not be made to coincide with the triedral angle whose vertex is T, it may be made to coincide with its symmetrical triedral angle, and the corresponding plane angles would be equal, as before.

NOTE 1. The projection of a point on a plane is the foot of a perpendicular drawn from the point to the plane.

NOTE 2.-The projection of a line on a plane is that line of the plane which joins the projection of the two extreme points of the given line on the plane.

EXERCISES.

1. Find a point in a plane equidistant from two given points without and on the same side of the plane.

2. From two given points on the same side of a given plane, draw two lines that shall meet the plane in the same point and make equal angles with it.

[The angle made by a line with a plane is the angle which the line makes with its projection on the plane.]

3. What is the greatest number of equilateral triangles that can be grouped about a point so as to form a convex polyedral angle?

4. Show that if from any two points in the edge of a diedral angle straight lines are drawn in each of its faces perpendicular to the edge, these lines contain equal angles.

5. From any point within a diedral angle, draw a perpendicular to each of its two faces, and show that the angle contained by the perpendiculars is the supplement of the diedral angle.

6. Show that if a plane meets another plane, the sum of the adjacent diedral angles is equal to two right angles.

7. Show that if two planes intersect each other, the opposite or vertical diedral angles are equal to each other. 8. Show that if a plane intersects two parallel planes, the sum of the interior diedral angles on the same side is equal to two right angles.

9. Show that if two diedral angles have their faces parallel and lying in the same or in opposite directions, they are equal.

10. Show that every point of a plane bisecting a diedral angle is equidistant from the faces of the angle.

11. Show that the inclination of a line to a planethat is, the angle which the line makes with its own projection on the plane is the least angle made by the line with any line of the plane.

12. Show that if three lines are perpendicular to a fourth at the same point, the first three are in the same plane.

13. Show that when a plane is perpendicular to a given line at its middle point, every point of the plane is equally distant from the extremities of the line, and that every point out of the plane is unequally distant from the extremities of the line.

14. Show that through a line parallel to line parallel to a given plane, but one plane can be passed perpendicular to the given plane.

15. Show that if two planes which intersect contain two lines parallel to each other, the intersection of the planes is parallel to the lines.

16. Show that when a line is parallel to one plane and perpendicular to another, the two planes are perpendicular to each other.

17. Draw a perpendicular to two lines not in the same plane.

18. Show that the three planes which bisect the diedral angles formed by the consecutive faces of a triedral angle, meet in the same line.