ABCD is a rhombus such that one of its diagonals is equal to its side. Find the measure of the angles of rhombus ABCD.

In a rhombus, the diagonals are perpendicular bisectors of each other. Let's denote the diagonal as 'd' and the side as 'a'. Given that d = a, we can use the Pythagorean theorem to find the angles.

Step 1: In triangle ABD (or BCD), since AB = AD = d (given) and BD = d (since BD is a diagonal), triangle ABD is an isosceles triangle.

Step 2: Let's denote the angle ∠BAD as 'x'. In triangle ABD, ∠ABD = ∠ADB = x (since it's an isosceles triangle).

Step 3: The sum of the angles in a triangle is 180 degrees. So, ∠BDA = 180 - 2x.

Step 4: Since the diagonals of a rhombus are perpendicular, ∠BDA = 90 degrees.

Step 5: Setting the equations from step 3 and step 4 equal to each other, we get 180 - 2x = 90. Solving for 'x' gives x = 45 degrees.

So, the measure of the angles of rhombus ABCD are 45 degrees and 135 degrees (since the angles of a rhombus add up to 180 degrees).