1. **State the problem:** We have two triangles with given angles and need to find the unknown angles $x$, $y$, and $z$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always $180$ degrees. 3. **First triangle angles:** $x$, $45$, and $66$ degrees. Apply the angle sum rule: $$x + 45 + 66 = 180$$ 4. **Solve for $x$:** $$x + 111 = 180$$ $$x = 180 - 111 = 69$$ 5. **Second triangle angles:** $30$, $y$, and $43$ degrees. Apply the angle sum rule: $$30 + y + 43 = 180$$ 6. **Solve for $y$:** $$y + 73 = 180$$ $$y = 180 - 73 = 107$$ 7. **Find $z$:** The angle $z$ and the given $119$ degrees are supplementary because they form a straight line. Use the supplementary angle rule: $$z + 119 = 180$$ 8. **Solve for $z$:** $$z = 180 - 119 = 61$$ **Final answers:** $$x = 69, y = 107, z = 61$$