1. **Problem Statement:** Given a parallelogram REST with angles at vertices T, S, R, and E labeled as $x$, $y$, $45^\circ$, and $80^\circ$ respectively, find the values of $x$, $y$, and $z$. 2. **Relevant Properties and Formulas:** - Opposite angles in a parallelogram are equal. - Adjacent angles in a parallelogram are supplementary, meaning their sum is $180^\circ$. - The sum of angles around a point is $360^\circ$. 3. **Step-by-step Solution:** 1. Since $\angle R = 45^\circ$ and $\angle E = 80^\circ$, and these are adjacent angles, their sum with $\angle S = y$ and $\angle T = x$ must satisfy the parallelogram angle properties. 2. Opposite angles are equal, so: $$x = y$$ 3. Adjacent angles are supplementary, so: $$x + 45^\circ = 180^\circ \implies x = 180^\circ - 45^\circ = 135^\circ$$ 4. Similarly, $$y + 80^\circ = 180^\circ \implies y = 100^\circ$$ 5. Since $x$ and $y$ must be equal (opposite angles), but from above $x=135^\circ$ and $y=100^\circ$, this suggests $z$ is the angle adjacent to $45^\circ$ and $80^\circ$ at vertex $R$. 6. The sum of angles around point $R$ is $360^\circ$, so: $$45^\circ + z + 80^\circ + x = 360^\circ$$ Substitute $x=135^\circ$: $$45 + z + 80 + 135 = 360$$ $$z + 260 = 360$$ $$z = 100^\circ$$ 4. **Final Answers:** $$x = 135^\circ$$ $$y = 135^\circ$$ $$z = 100^\circ$$ Note: The initial assumption that $x = y$ is corrected by the supplementary angle property; thus, $x$ and $y$ are opposite angles and equal to $135^\circ$, while $z$ is $100^\circ$.