Let's analyze the given quadrilateral so that we can find the measure of ∠ZXY.
First, we must find the value of x. By the definition of a rectangle, we know that WXYZ has four right angles. Therefore, the measure of m ∠WXY is 90.
m ∠WXY= 90With the Angle Addition Postulate we can express $m \angle WXY$ as a sum of $m \angle ZXY$ and $m \angle ZXW.$
\begin{gathered}
m \angle ZXY + m \angle ZXW = m \angle WXY \\
\Updownarrow \\
m \angle ZXY + m \angle ZXW = \colV{90}
\end{gathered}
We are given that $m \angle XZW=\colVI{x-11}.$ We will substitute this expressions into our equation.
\begin{gathered}
m \angle ZXY + \colVI{x-11} = 90
\end{gathered}
Because our quadrilateral is a rectangle, both pairs of opposite sides are parallel. Recall the following theorem.
Alternate Interior Angles Theorem
If parallel lines are cut by a transversal, then the pair of alternate interior angles are congruent.
Notice that by this theorem, $\angle ZXY$ and $\angle WZX$ are alternate interior angles, with $\seg{ZX}$ as the transversal. Therefore, $\angle ZXY$ and $\angle WZX$ are congruent. An expression for the measure of $\angle WZX$ is given as $\colV{x-9},$ and we can substitute it into the expression.
\begin{aligned}
m \angle ZXY&=m \angle WZX \ \\ &\Updownarrow \\ m \angle ZXY&= \colIV{x-9}
\end{aligned}
Now that we have an expression for the measure of $\angle WZX$, let's substitute it back into the previous equation.
\begin{gathered}
\colIV{x-9} + \colVI{x-11} = \colV{90}
\end{gathered}
Let's solve it!
