1. **State the problem:** We have a quadrilateral with angles labeled as $ (8x - 7)^\circ $, $ (7x + 2)^\circ $, $ (10x)^\circ $, and $ 165^\circ $. We need to find the value of $x$. 2. **Formula used:** The sum of the interior angles of any quadrilateral is $360^\circ$. 3. **Set up the equation:** $$ (8x - 7) + (7x + 2) + 10x + 165 = 360 $$ 4. **Simplify the equation:** $$ 8x - 7 + 7x + 2 + 10x + 165 = 360 $$ $$ (8x + 7x + 10x) + (-7 + 2 + 165) = 360 $$ $$ 25x + 160 = 360 $$ 5. **Isolate $x$:** $$ 25x + 160 = 360 $$ $$ 25x = 360 - 160 $$ $$ 25x = 200 $$ 6. **Divide both sides by 25:** $$ x = \frac{200}{25} $$ $$ x = \cancel{\frac{200}{25}} \Rightarrow 8 $$ 7. **Final answer:** $$ \boxed{8} $$ This means the value of $x$ that satisfies the angle measures in the quadrilateral is 8.