The measure of angle BCD in a rhombus is found using the properties of the rhombus, specifically the bisected angles, and is calculated to be 36 degrees.
To solve the problem involving the measure of angle BCD in a rhombus, we need to understand the properties of a rhombus. Since the diagonals of a rhombus bisect each other at right angles, angle BAE and angle ABE form a linear pair with angle AED, which means their measures add up to 90 degrees. Given that measure of angle BAE is ⅓ (the measure of angle ABE), we can use the equation BAE + ABE = AED = 90 degrees to find the measure of these angles.
If we let the measure of angle ABE be 3x, then the measure of angle BAE would be 2x, and setting up the equation 2x + 3x = 90 leads to 5x = 90 and x = 18 degrees. Therefore, angle ABE is 54 degrees (3x), and angle BAE is 36 degrees (2x). The measure of angle ABC, which is adjacent to angle ABE, will be the same as angle ABE because opposite angles in a rhombus are congruent. Thus, angle ABC is 54 degrees. Since the diagonals bisect each other at right angles, angle CBD is 90 degrees.
The measure of angle BCD is the complementary angle to angle ABC within the triangle ABC. So, angle BCD is 180 degrees - 90 degrees - angle ABC, which is 180 degrees - 90 degrees - 54 degrees = 36 degrees.
In conclusion, the measure of angle BCD is 36 degrees.