1. **State the problem:** We have pairs of right triangles with angles labeled as $(y + 4)^\circ$ and $(2y - 15)^\circ$ and sides involving expressions with $x$. We want to find the values of $x$ and $y$ that satisfy the angle relationships. 2. **Use the angle sum property of triangles:** The sum of angles in any triangle is $180^\circ$. Since these are right triangles, one angle is $90^\circ$. Therefore, the other two angles must sum to $90^\circ$: $$ (y + 4) + (2y - 15) = 90 $$ 3. **Simplify the equation:** $$ y + 4 + 2y - 15 = 90 $$ $$ 3y - 11 = 90 $$ 4. **Solve for $y$:** $$ 3y = 90 + 11 $$ $$ 3y = 101 $$ $$ y = \frac{101}{3} $$ 5. **Calculate $y$ value:** $$ y = 33.666\ldots \approx 33.67 $$ 6. **Use trigonometric ratios to find $x$:** For the triangle with sides 12 (vertical) and 5 (horizontal), and angle $(y + 4)^\circ$, use tangent: $$ \tan(y + 4) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5} $$ 7. **Calculate $\tan(y + 4)$:** $$ y + 4 = 33.67 + 4 = 37.67^\circ $$ $$ \tan(37.67^\circ) \approx 0.77 $$ 8. **Check consistency:** Since $\frac{12}{5} = 2.4$, and $\tan(37.67^\circ) \approx 0.77$, this suggests the angle labeling or side assignment might differ. Instead, use the other angle $(2y - 15)^\circ$ and side expressions to find $x$. 9. **Use the right triangle with horizontal side $(2x - 3)$ and angle $(2y - 15)^\circ$:** Since the vertical side is 12, use tangent: $$ \tan(2y - 15) = \frac{12}{2x - 3} $$ 10. **Calculate $2y - 15$:** $$ 2y - 15 = 2 \times 33.67 - 15 = 67.34 - 15 = 52.34^\circ $$ 11. **Calculate $\tan(52.34^\circ)$:** $$ \tan(52.34^\circ) \approx 1.28 $$ 12. **Set up equation and solve for $x$:** $$ 1.28 = \frac{12}{2x - 3} $$ Multiply both sides by $2x - 3$: $$ 1.28(2x - 3) = 12 $$ $$ 2.56x - 3.84 = 12 $$ Add $3.84$ to both sides: $$ 2.56x = 15.84 $$ Divide both sides by $2.56$: $$ x = \frac{15.84}{2.56} $$ 13. **Calculate $x$ value:** $$ x = 6.1875 $$ **Final answers:** $$ y = \frac{101}{3} \approx 33.67 $$ $$ x = 6.19 $$