Let the measure of the angle be \( x \) degrees. The complementary angle, which is the angle that adds up to 90° with \( x \), can be expressed as \( 90° - x \).
According to the problem, the angle \( x \) measures 60° less than its complementary angle:
\[ x = (90° - x) - 60° \]
Now, we can simplify the equation:
\[ x = 90° - x - 60° \] \[ x = 30° - x \]
Next, we add \( x \) to both sides to combine like terms:
\[ x + x = 30° \] \[ 2x = 30° \]
Now, divide both sides by 2:
\[ x = 15° \]
Now we have found the measure of the angle. To find the measure of the complementary angle, we substitute \( x \) back into the expression for the complementary angle:
\[ 90° - x = 90° - 15° = 75° \]
Thus, the measures of the two angles are:
- The angle: \( 15° \)
- The complementary angle: \( 75° \)
To summarize, the measures of the angles are \( 15° \) and \( 75° \).