答案：E

解析：

Triangle A B C is isosceles, and the measure of angle A is 74 degrees. Let the measure of angle B be b degrees, and let the measure of angle C be c degrees. The sum of the measures of the angles of a triangle is 180 degrees, so 74 plus b plus c equals 180. An isosceles triangle has two angles of equal measure, so b equals 74, or c equals 74, or b equals c does not equal 74.

If b equals 74, then 74 plus 74 plus c equals 180, so c equals 32, and the measures of the angles of the triangle are 74 degrees, 74 degrees and 32 degrees.

If c equals 74, then 74 plus b plus 74 equals 180, so b equals 32, and the measures of the angles of the triangle are 74 degrees, 32 degrees and 74 degrees.

If b equals c, then 74 plus b plus c equals 74 plus (2 times b) equals 180, so 2 times b equals 106, and b equals c equals 53. In this case, the measures of the angles of the triangle are 74 degrees, 53 degreesand 53 degrees.

Hence if the measure of angle A is 74 degrees, the possible measures of angles B and C, respectively, are 74 degrees and 32 degrees; 32 degrees and 74 degrees; and 53 degrees and 53 degrees. Therefore, it is not possible to determine what must be the measure of another angle of this triangle.