it. Thus, in the quarter from 0 to 90, there are 90 degrees, as marked;
but suppose the diameter of the circle were that of inner circle _d_,
and one-quarter of it would still contain 90 degrees.

[Illustration: Fig. 56.]

So, likewise, the degrees of one line to another are not always taken
from one point, as from the point O, but from any one line to another.
Thus the line marked 120 is 60 degrees from line 180, or line 90 is 60
degrees from line 150. Similarly in the other quarter of the circle 60
degrees are marked. This may be explained further by stating that the
point O or zero may be situated at the point from which the degrees of
angle are to be taken. Here it may be remarked that, to save writing the
word "degrees," it is usual to place on the right and above the figures
a small deg., as is done in Figure 56, the 60 deg. meaning sixty degrees, the
deg., of course, standing for degrees.

[Illustration: Fig. 57.]

Suppose, then, we are given two lines, as _a_ and _b_ in Figure 57, and
are required to find their angle one to the other. Then, if we have a
protractor, we may apply it to the lines and see how many degrees of
angle they contain. This word "contain" means how many degrees of angle
there are between the lines, which, in the absence of a protractor, we
may find by prolonging the lines until they meet in a point as at _c_.
From this point as a centre we draw a circle D, passing through both
lines _a_, _b_. All we now have to do is to find what part, or how much
of the circumference, of the circle is enclosed within the two lines. In
the example we find it is the one-twelfth part; hence the lines are 30
degrees apart, for, as the whole circle contains 360, then one-twelfth
must contain 30, because 360/12 = 30.

[Illustration: Fig. 58.]

If we have three lines, as lines A B and C in Figure 58, we may find
their angles one to the other by projecting or prolonging the lines
until they meet as at points D, E, and F, and use these points as the
centres wherefrom to mark circles as G, H, and I. Then, from circle H,
we may, by dividing it, obtain the angle of A to B or of B to A. By
dividing circle I we may obtain the angle of A to C or of C to A, and by
dividing circle G we may obtain the angle of B to C or of C to B.

[Illustration: Fig. 59.]

It may happen, and, indeed, generally will do so, that the first attempt
will not succeed, because the distance between the lines measured, or
the arc of th