The sides of a triangle measure 14.9 centimeters, 23.8 centimeters, and 36.9 centimeters. find the measure of the angle with the least measure.

It is convenient to let a triangle solver figure this one.
  The smallest angle is about 13.8°.

If you want to do it by hand, you can make use of the law of cosines.
  a² = b² + c² - 2bc·cos(A)
Solving for the angle, you have
  A = arccos((b²+c²-a²)/(2bc))

Substituting the given values gives you
  A = arccos((23.8²+36.9²-14.9²)/(2·23.8·36.9))
  A = arccos(1706.04/1756.44)
  A ≈ arccos(0.9713056)
  A ≈ 13.75879° ≈ 13.8°