Since vertical angles are equal, we can denote the measures of the two vertical angles as \( x \) and \( x \) (since they are the same).
Given that the sum of the two angles is 300 degrees, we can set up the equation:
\[ x + x = 300 \]
This simplifies to:
\[ 2x = 300 \]
Now, we divide both sides by 2:
\[ x = 150 \]
Thus, the measure of each angle is 150 degrees.
So the answer is 150.