1. The problem states that $\angle 1$ and $\angle 2$ are vertical angles, and we are given $m\angle 1 = 7x + 1$ and $m\angle 2 = 5x + 47$. We need to find $m\angle 2$. 2. Vertical angles are always congruent, meaning their measures are equal. So, we set the expressions equal: $$7x + 1 = 5x + 47$$ 3. Solve for $x$: Subtract $5x$ from both sides: $$7x - 5x + 1 = 47$$ Simplify: $$2x + 1 = 47$$ Subtract 1 from both sides: $$2x = 46$$ Divide both sides by 2: $$x = 23$$ 4. Substitute $x = 23$ back into the expression for $m\angle 2$: $$m\angle 2 = 5(23) + 47 = 115 + 47 = 162$$ 5. Therefore, the measure of $\angle 2$ is $162^\circ$.