1. **State the problem:** We are given two angles expressed in terms of $x$: $4x - 1$ degrees and $2x + 5$ degrees, and we need to find the measure of angle $m\angle MYW$. The diagram shows right angles, indicating relationships between these angles. 2. **Analyze the figure and given information:** The two right angle marks indicate that the angles adjacent to $m\angle MYW$ are $90^\circ$ each. Since rays $P$, $M$, and $W$ originate from point $Y$, the sum of angles around point $Y$ on a straight line is $180^\circ$. 3. **Set up the equation:** The sum of the two given angles plus $m\angle MYW$ equals $90^\circ$ because the right angles split the space. So, $$ (4x - 1) + (2x + 5) + m\angle MYW = 90 $$ 4. **Simplify the equation:** $$ 4x - 1 + 2x + 5 + m\angle MYW = 90 $$ $$ 6x + 4 + m\angle MYW = 90 $$ 5. **Express $m\angle MYW$ in terms of $x$:** $$ m\angle MYW = 90 - 6x - 4 $$ $$ m\angle MYW = 86 - 6x $$ 6. **Use the given value $m\angle MYW = \frac{43}{3}$ degrees:** $$ 86 - 6x = \frac{43}{3} $$ 7. **Solve for $x$:** Multiply both sides by 3 to clear the fraction: $$ 3(86 - 6x) = 43 $$ $$ 258 - 18x = 43 $$ Subtract 258 from both sides: $$ -18x = 43 - 258 $$ $$ -18x = -215 $$ Divide both sides by $-18$: $$ x = \frac{-215}{\cancel{-18}} \cancel{\div -1} = \frac{215}{18} $$ 8. **Calculate $m\angle MYW$ using $x = \frac{215}{18}$:** $$ m\angle MYW = 86 - 6 \times \frac{215}{18} $$ $$ m\angle MYW = 86 - \frac{1290}{18} $$ Simplify $\frac{1290}{18}$: $$ \frac{1290}{18} = 71.666\ldots = \frac{215}{3} $$ So, $$ m\angle MYW = 86 - \frac{215}{3} = \frac{258}{3} - \frac{215}{3} = \frac{43}{3} $$ **Final answer:** $$ m\angle MYW = \frac{43}{3} \text{ degrees} $$ This matches the given value, confirming the solution is consistent.