Final answer:
The diagonals of a rhombus divide it into four congruent right triangles. By applying the Pythagorean theorem to these triangles, we find out that the rhombus has angles of 60° and 120°, which are not represented in any of the given options.
Step-by-step explanation:
To determine the measures of the angles of the rhombus with diagonals of length a and a√3 (a multiplied by the square root of 3), we recall that the diagonals of a rhombus bisect each other at right angles and bisect the angles of the rhombus. Thus, each diagonal will cut the rhombus into four congruent right triangles. By using the Pythagorean theorem we can find the relation between the sides of these right triangles.
Let's consider one of these right triangles formed by the diagonals. The lengths of this right triangle's legs are half the lengths of the diagonals of the rhombus, so they are ½a and ½a√3. Using the Pythagorean theorem, given by a² + b² = c², we can see that (½a)² + (½a√3)² = ¼(a²) + ¼(3a²) = a², indicating that the hypotenuse of the right triangle is a. Since we have a 30-60-90 right triangle, the angles of the rhombus are twice the angles of this right triangle, resulting in 60° and 120°. Therefore, the angles of the rhombus are 60°, 120°, 60°, and 120°, making none of the given options correct.