1. **State the problem:** We need to find the values of $x$, $y$, and $z$ in a rhombus where the angles are given as $(-x+1)^\circ$, $(3z+9)^\circ$, $108^\circ$, and $(-2y+4)^\circ$. 2. **Recall properties of a rhombus:** All sides are equal, and opposite angles are equal. Also, the sum of all interior angles is $360^\circ$. 3. **Set up equations:** - Opposite angles are equal: $$-x+1 = 108$$ $$3z+9 = -2y+4$$ - Sum of all angles: $$(-x+1) + (3z+9) + 108 + (-2y+4) = 360$$ 4. **Solve for $x$ from the first equation:** $$-x + 1 = 108$$ $$-x = 108 - 1$$ $$-x = 107$$ $$\cancel{-x} = \cancel{107}$$ $$x = -107$$ 5. **Express $3z+9$ in terms of $y$ from the second equation:** $$3z + 9 = -2y + 4$$ $$3z = -2y + 4 - 9$$ $$3z = -2y - 5$$ 6. **Substitute all angles into the sum equation:** $$(-x+1) + (3z+9) + 108 + (-2y+4) = 360$$ Substitute $x = -107$: $$(-(-107) + 1) + (3z + 9) + 108 + (-2y + 4) = 360$$ $$ (107 + 1) + (3z + 9) + 108 + (-2y + 4) = 360$$ $$108 + 3z + 9 + 108 - 2y + 4 = 360$$ $$ (108 + 9 + 108 + 4) + 3z - 2y = 360$$ $$229 + 3z - 2y = 360$$ 7. **Simplify:** $$3z - 2y = 360 - 229$$ $$3z - 2y = 131$$ 8. **Recall from step 5:** $$3z = -2y - 5$$ 9. **Substitute $3z$ from step 8 into step 7:** $$(-2y - 5) - 2y = 131$$ $$-2y - 5 - 2y = 131$$ $$-4y - 5 = 131$$ $$-4y = 131 + 5$$ $$-4y = 136$$ $$\cancel{-4y} = \cancel{136}$$ $$y = -34$$ 10. **Find $z$ using $3z = -2y - 5$:** $$3z = -2(-34) - 5$$ $$3z = 68 - 5$$ $$3z = 63$$ $$\cancel{3z} = \cancel{63}$$ $$z = 21$$ **Final answers:** $$x = -107, \quad y = -34, \quad z = 21$$