to CD and it will be a perpendicular to xy, by our proposition; and it must therefore coincide with the perpendicular BA (13). Since then, BA coincides with a parallel to DC, BA must be parallel to DC.

25. COROLLARY 2. Parallel planes are everywhere equally distant.

Consider the parallel planes MN and PQ, of PROP. V. Let AA' and CC' be any two perpendiculars to the plane PQ between the parallel planes; then will AA' and CC' be equal.

For, AA' is parallel to CC' (24),
Hence, AA' is equal to CC' (17).

PROPOSITION VIII.

26. THEOREM. Straight lines cut by three parallel planes are divided proportionally.

Let the lines AC and A'C' be cut by the parallel planes MN, PQ, and RS, in the points A, B, C, and A', B', C'; then will

AB: BC: A'B' : B'C'.

Draw the line AC' piercing PQ at D; then draw BD and DB', also draw AA' and CC'. Now BD is parallel to CC', and DB' is parallel to AA' (18).

EXERCISES.

27. THEOREM. If A be any point without a plane: (a). The perpendicular from A to the plane is shorter than any oblique line.

(b). Oblique lines from A cutting off equal distances from the foot of the perpendicular are equal.

(c). Of two oblique lines cutting off unequal distances from the foot of the perpendicular, the more remote is the greater.

Let the line AB be drawn from A, perpendicular to the plane MN, meeting it at any point B. Let the points C and D be taken in MN, equally distant from B; and let E be a point more remote.

28. A diedral angle is the angle between two planes which intersect each other.

29. The line in which the planes intersect is called the edge of the angle; the planes themselves are called the faces of the angle.

30. Lines drawn in the faces of a diedral angle perpendicular to the edge and from the same point in it, form a plane angle which is taken as the measure of the diedral angle.

D

A

If the plane DE intersect the plane DF in the line DG, then F-GD-E, the angle between the planes,

A

F

B

B'

E

is a diedral angle. DF and DE are faces and DG is the edge of the diedral angle.

In the face DF draw the line AC perpendicular to the edge GD at the point A. And, in the face DE draw the line AB perpendicular to the edge GD at the point A.

Then CAB, the plane angle between the lines CA and AB, is the measure of the diedral angle F-GD-E.

For, if we revolve DF about DG as an axis, the angle CAB will increase or diminish precisely as the angle F-GD-E increases or diminishes. And, since this is not the case with any other plane angle whose vertex is at A and whose sides are in DF and DE respectively, CAB is taken as the measure of F-GD-E.

Construct a second plane angle C'A'B', as before; then CAB is equal to C'A'B' . . . (19).

PROPOSITION IX.

31. THEOREM. If a line is perpendicular to a plane, any plane passing through the line is perpendicular to the plane.

Let the line BA be perpendicular to the plane MN; and let the plane PQ be passed through BA; then will PQ be perpendicular to MN.

In MN draw the line BS perpendicular to BQ, the intersection of PQ and MN. Now since AB and BS lying in their respective planes PQ and MN, are perpendicular to

the intersection of these planes, at a common point, the angle ABS is the measure of the diedral angle between the planes PQ and MN ... (30). But since ABS is a right angle (4), PQ is perpendicular to MN. Q. E. D.

32. COROLLARY 1. A line drawn in one of two perpendicular planes, perpendicular to their intersection, is perpendicular to the other plane.

Thus, as we see, the line BA drawn in the plane PQ perpendicular to the intersection BQ, is perpendicular to the plane MN.

33. COROLLARY 2. If a line be drawn perpendicular to one of two perpendicular planes, at a point of their intersection, it will lie in the other plane.

Thus, as we see, the line BA drawn perpendicular to the plane MN, at B in the intersection, lies in the plane PQ perpendicular to MN.

34. COROLLARY 3. If two intersecting planes are perpendicular to a third plane, their line of intersection will also be perpendicular to the third plane.

Thus, as we see, the line BA drawn perpendicular to MN, is the intersection of the planes PQ and RS passing through BA (12), which planes are perpendicular to MN (31).

DEFINITIONS.

35. A polyedral angle is an angle formed by three or more planes meeting at a common point.

36. The point in which the planes meet is called the vertex of the angle. The lines in which the planes meet are called the edges of the angle. Those portions of planes which lie between the edges are termed the faces.

37. The plane angles lying between the edges and having a common vertex, are called face angles.

38. The face angles and the diedral angles between the faces are called the parts of the polyedral angle.

39. Two polyedral angles in which all the parts of the one are equal to all the parts of the other, each to each, are said to be mutually equal in all their parts, or, with respect to all their parts.

B

C A

FIG. 1.

B'

40. If the equal parts of the one are arranged in the same order as the equal parts of the other, throughout, the polyedral angles are superposable, and hence equal. See Fig. 1.

41. If the equal parts of the one are arranged in the reverse order of the equal parts of the other, the polyedral angles are evidently not superposable; in this case they are symmetrical. See Fig. 2.

solids will then be in the relative position of an object standing upon a horizontal mirror and the image of the object.

"A familiar example of the symmetry of non-superposable figures is afforded by a pair of outstretched hands. They can be so placed, palm opposite to palm, that each is the image of the other."

43. A triedral angle is a polyedral angle of three faces.