Elements of descriptive geometry, with applications to isometric projection and othe forms of one-plane projection; a text-book for colleges and ingineering schools by O. E. Randall.

RELATING TO POINT, LINE, AND PLANE

67

The oblique angle at the base of the triangle is the required
angle, and since the triangle is right-angled the angle at the
vertex must be the complement of the required angle.
Construction. See Fig. 76. Let S represent the given plane and
let MI-N represent the given line.
Following Analysis 2, through some point of M-N, as M,
draw M1-B perpendicular to S and produce it to intersect H at b,.
Find the point a, in which M-N intersects H. The line a,-b,
is the horizontal trace of the
plane of the two lines IM-N
and M-B. X /
Revolve this plane about mc
a,b, as an axis, into H. The -
points a, and b, remain station- -.
ary and M revolves to mH. /! '
The angle a,-mi-b, is the true G b a L
size of the angle between the n
given line and the line perpen, /
dicular to the plane, and is,
by analysis, the complement of
the required angle. FIG. 76
Draw mH-eH perpendicular to mH-b,. The angle a,-mH-eH, which
by construction is the complement of the angle a,-m,-b,, is the
required angle.
Check. Solve by Analysis 1, or assume some point other than
M, in M-N7, as the point through which to draw the line perpendicular to the plane.
186. Problem 124. Find the angle between the line [1M= - 6, 3, 5;
N= 5, -4, 2] and the plane S= 0~, 4, 6.
187. Problem 125. Find the angle between the line [M= - 6, 3, 5;
N= 5, -4, 1] and the plane S = 0, 90~, 90~.
188. Problem 126. Find the angle between the line [M= 0, 6, - 1;
N= 0, - 5, - 6] and the plane S = 0~, 6, 3.
189. Problem 127. To find the angle between two given planes.
A dihedral angle is measured by the plane angle formed by two
straight lines perpendicular to the edge at the same point, one line
lying in one face and the other line lying in the other face.