1. **Stating the problem:** We need to find the sizes of the unknown angles $a$, $b$, $c$, $d$, and $e$ in the given triangle diagram, where some angles and parallel lines are given. 2. **Important rules and formulas:** - The sum of angles in any triangle is $180^\circ$. - Alternate interior angles are equal when lines are parallel. - Corresponding angles are equal when lines are parallel. 3. **Given angles:** - $\angle U = 54^\circ$ - $\angle X = 50^\circ$ 4. **Find $d$ at vertex $T$:** Since $TU$ is parallel to $ZX$ (from parallel marks), and $\angle X = 50^\circ$ is alternate interior to $\angle d$, we have: $$d = 50^\circ$$ 5. **Find $a$ at vertex $W$:** Since $WU$ is parallel to $VX$, and $\angle U = 54^\circ$ is alternate interior to $\angle a$, we have: $$a = 54^\circ$$ 6. **Find $e$ at vertex $X$:** $e$ and $\angle X = 50^\circ$ are on a straight line (linear pair), so: $$e + 50^\circ = 180^\circ \implies e = 130^\circ$$ 7. **Find $c$ at vertex $X$:** Since $c$ and $e$ are angles around point $X$ on a straight line, and $e = 130^\circ$, then: $$c = 180^\circ - e = 50^\circ$$ 8. **Find $b$ at vertex $V$:** In triangle $ZVX$, sum of angles is $180^\circ$: $$b + c + \angle X = 180^\circ$$ Substitute known values: $$b + 50^\circ + 50^\circ = 180^\circ$$ $$b = 180^\circ - 100^\circ = 80^\circ$$ **Final answers:** $$a = 54^\circ, b = 80^\circ, c = 50^\circ, d = 50^\circ, e = 130^\circ$$