In the triangle below, determine the measure of angle C

http://tinypic.com/r/2a4oapx/6

You can use the law of sines to find angle C. But you'll need the side of C as well. You can get that by using the Pythagorean Theorem, which results in sqrt(176). With the law of sines:
(sin C)/sqrt(176) = (sin 90)/15

Cross-multiply and solve:
15sin C = sqrt(176)sin 90
15sin C = sqrt(176)
sin C = sqrt(176)/15
C = ~62.182 degrees

EDIT: I did it the long way, but Josh K's answer works as well.

by pythagoras' theory,

AB^2 + BC^2 = AC^2

AB^2 = AC^2 - BC^2

= 225 - 49

AB^2 = 176

AB = sqrt176

we can now use the sine rule,

sin B/CA = sinC/AB

sin90/15 = sinC/sqrt176

1/15 = sinC/sqrt176

sinC = sqrt176/15

C = sin^(-1)[(sqrt176)/15]

C = cos^(-1)[7/15]

Right?

isn't cosC = 7/15 by the picture, and cos = A / H

Then C = cos^(-1)[7/15]