1. **State the problem:** We need to find the measure of the unknown angle $x$ given three angles expressed in terms of $x$: $x + 15^\circ$, $3x - 22^\circ$, and $2x - 10^\circ$. 2. **Understand the geometric context:** The angles are part of a polygon with points $u, w, x, y, z$. The angles at $w$, $x$, and $y$ are given, and points $y$ and $z$ form right angles. Since the figure is polygonal, the sum of the interior angles depends on the number of sides. 3. **Sum of interior angles formula:** For a polygon with $n$ sides, the sum of interior angles is $$180^\circ \times (n - 2).$$ 4. **Determine the number of sides:** The polygon has points $u, w, x, y, z$, so $n=5$. 5. **Calculate the sum of interior angles:** $$180^\circ \times (5 - 2) = 180^\circ \times 3 = 540^\circ.$$ 6. **Use the right angles at $y$ and $z$:** Each right angle is $90^\circ$, so angles at $y$ and $z$ contribute $90^\circ$ each. 7. **Set up the equation:** Sum of all interior angles = sum of given angles + right angles $$ (x + 15) + (3x - 22) + (2x - 10) + 90 + 90 = 540 $$ 8. **Simplify the equation:** $$ x + 15 + 3x - 22 + 2x - 10 + 180 = 540 $$ $$ (x + 3x + 2x) + (15 - 22 - 10 + 180) = 540 $$ $$ 6x + 163 = 540 $$ 9. **Solve for $x$:** $$ 6x = 540 - 163 $$ $$ 6x = 377 $$ $$ x = \frac{377}{6} = 62.8333^\circ $$ 10. **Find each angle measure:** - $x + 15 = 62.8333 + 15 = 77.8333^\circ$ - $3x - 22 = 3 \times 62.8333 - 22 = 188.5 - 22 = 166.5^\circ$ - $2x - 10 = 2 \times 62.8333 - 10 = 125.6666 - 10 = 115.6666^\circ$ **Final answer:** $$x = 62.83^\circ, \quad x + 15 = 77.83^\circ, \quad 3x - 22 = 166.5^\circ, \quad 2x - 10 = 115.67^\circ.$$