What is the measure of angle A in triangle ABC if B equals 15 degrees C equals 65 degrees and side a equals 19?

If you are really talking about a closed triangle ABC, then the length of side "a" (given as 19) does not matter in the calculation.

Sum of the angles of a triangle is 180 degrees. Angle B and C add up to 15 + 65 = 80 degrees. Hence angle A is (180 - 80) = 100 degrees