1. **State the problem:** We are given a quadrilateral with vertices labeled $\pi$, $x$, $y$, and $z$. The angles at vertices $x$ and $y$ are given as $(13x - 7)^\circ$ and $(8x - 2)^\circ$ respectively. Sides $\pi-x$ and $z-y$ are parallel. We need to find the measures of angles $\pi$, $x$, $y$, and $z$. 2. **Identify the properties and formulas:** Since $\pi-x$ is parallel to $z-y$, the consecutive interior angles along the transversal are supplementary. This means: $$m\angle x + m\angle y = 180^\circ$$ Also, the sum of interior angles in any quadrilateral is: $$m\angle \pi + m\angle x + m\angle y + m\angle z = 360^\circ$$ 3. **Set up the equation for $x$ and $y$:** $$ (13x - 7) + (8x - 2) = 180 $$ Simplify: $$ 13x - 7 + 8x - 2 = 180 $$ $$ 21x - 9 = 180 $$ Add 9 to both sides: $$ 21x - \cancel{9} + 9 = 180 + 9 $$ $$ 21x = 189 $$ Divide both sides by 21: $$ \frac{21x}{\cancel{21}} = \frac{189}{21} $$ $$ x = 9 $$ 4. **Calculate the measures of angles $x$ and $y$:** $$ m\angle x = 13(9) - 7 = 117 - 7 = 110^\circ $$ $$ m\angle y = 8(9) - 2 = 72 - 2 = 70^\circ $$ 5. **Find the measures of angles $\pi$ and $z$:** Since $\pi-x$ is parallel to $z-y$, angles $\pi$ and $z$ are also supplementary to $x$ and $y$ respectively, or we can use the quadrilateral angle sum: $$ m\angle \pi + 110 + 70 + m\angle z = 360 $$ $$ m\angle \pi + m\angle z + 180 = 360 $$ $$ m\angle \pi + m\angle z = 180 $$ Without additional information, we can assume $m\angle \pi = m\angle z$ (common in parallelograms), so: $$ 2m\angle \pi = 180 $$ $$ m\angle \pi = 90^\circ $$ Therefore: $$ m\angle z = 90^\circ $$ **Final answers:** $$ m\angle \pi = 90^\circ $$ $$ m\angle x = 110^\circ $$ $$ m\angle y = 70^\circ $$ $$ m\angle z = 90^\circ $$