Final answer:
The Vertical Angles Theorem states that vertical angles are congruent. By using this theorem, we can find the measures of the given angles by solving an equation involving x.
Step-by-step explanation:
The theorem or postulate related to the measures of the angles in a pair is the Vertical Angles Theorem. According to this theorem, vertical angles are congruent, which means they have equal measures.
In this case, we have two vertical angles, ∠4 and ∠6. Using the Vertical Angles Theorem, we can say that m∠4 = m∠6. So, 60x + 30 = 9x + 12.
We can solve this equation to find the value of x, and then substitute it back to find the measure of each angle.