What is the measure of angle a to the nearest degree?

There are several ways to find the angle at A.

One is to realize the bisector of angle A will make a right angle with BC, creating two congruent triangles with short leg 2 and hypotenuse 7. Then the sine of the half-angle at A will be 2/7, so
  A = 2*arcsin(2/7)
  A ≈ 2*16.6015° = 33.203°

The measure of the angle at A to the nearest degree is
  33°

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Another way to find the angle is to use the Law of Cosines. After solving for the angle (A) in terms of sides a, b, c, you find
  A = arccos((b²+c²-a²)/(2bc)) = arccos(7²+7²-4²)/(2*7*7))
  A = arccos(82/98) ≈ 33°

And you can always use a triangle solver application.