To find the measure of the complement of the given angle, we can use the fact that two angles are complementary if their sum is 90°.

Given that $m \angle DOG = 60^\circ$, the complement $x$ can be found as follows:

$x = 90^\circ - 60^\circ = 30^\circ$

So, the measure of the complement is $30^\circ$.

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. What are supplementary angles and how do they differ from complementary angles?
2. How would you find the complement if the angle was given as a variable, say $x^\circ$?
3. Can an obtuse angle have a complement? Why or why not?
4. What is the relationship between complementary angles and right angles?
5. How can you verify if two angles are complementary?

Tip: Always remember that complementary angles sum up to 90°, while supplementary angles sum up to 180°.