1. **State the problem:** We have two parallel lines $a$ and $b$ cut by a transversal $t$. Given $m\angle 3 = 4x - 31$ and $m\angle 8 = 2x + 7$, find the value of $x$ (Question 14) and then find $m\angle 3$ (Question 15). 2. **Identify the relationship between angles:** Since lines $a$ and $b$ are parallel and $t$ is a transversal, angles 3 and 8 are alternate interior angles, which means they are equal: $$m\angle 3 = m\angle 8$$ 3. **Set up the equation:** $$4x - 31 = 2x + 7$$ 4. **Solve for $x$:** Subtract $2x$ from both sides: $$4x - \cancel{2x} - 31 = \cancel{2x} + 7 \implies 2x - 31 = 7$$ Add 31 to both sides: $$2x - 31 + 31 = 7 + 31 \implies 2x = 38$$ Divide both sides by 2: $$\frac{\cancel{2}x}{\cancel{2}} = \frac{38}{2} \implies x = 19$$ 5. **Find $m\angle 3$ using $x=19$:** $$m\angle 3 = 4x - 31 = 4(19) - 31 = 76 - 31 = 45$$ 6. **Check answer choices:** - For $x$, none of the options (24, 34, 44, 54) match 19, so re-check the angle relationship. 7. **Re-examine angle relationship:** Angles 3 and 8 are not alternate interior angles but are *supplementary* because they are consecutive interior angles on the same side of the transversal. So, $$m\angle 3 + m\angle 8 = 180$$ 8. **Set up the correct equation:** $$4x - 31 + 2x + 7 = 180$$ Simplify: $$6x - 24 = 180$$ Add 24 to both sides: $$6x = 204$$ Divide both sides by 6: $$\frac{\cancel{6}x}{\cancel{6}} = \frac{204}{6} \implies x = 34$$ 9. **Find $m\angle 3$ with $x=34$:** $$m\angle 3 = 4(34) - 31 = 136 - 31 = 105$$ 10. **Answer:** - For question 14, $x = 34$ (Option B). - For question 15, $m\angle 3 = 105$ (Option A).