Problem
13. If $\overrightarrow{B D} \perp \overrightarrow{A C}, m \angle D B E=(2 x-1)^{\circ}$, and $m \angle C B E=(5 x-42)^{\circ}$, find the value of $x$. 14. Find th $m \angle P Q$ P $Q$ 15. Find the values of $x$ and $y$.
Solution
Step 1 ：Given that $\overrightarrow{B D} \perp \overrightarrow{A C}$, $m \angle D B E=(2 x-1)^{\circ}$, and $m \angle C B E=(5 x-42)^{\circ}$, we know that $\angle DBE$ and $\angle CBE$ are supplementary angles, meaning their measures add up to 180 degrees.
Step 2 ：Set up the equation $(2x - 1) + (5x - 42) = 180$ and solve for $x$.
Step 3 ：Solving the equation gives $x = \frac{223}{7}$.
Step 4 ：Final Answer: The value of $x$ is $\boxed{\frac{223}{7}}$.
