Solution:

Let us use measuring units in which the side length of each key is $1$. We express the area of quadrilateral $Q A Z E$ in two ways:

a. By dividing into triangles $Q A Z$ and $Q Z E$. Since $\triangle Q Z E$ has base $Q E = 2$ and height $2$, we get
$$\text{Area } Q A Z E = \text{Area } Q A Z + \frac{1}{2} \cdot 2 \cdot 2 = \operatorname{Area} Q A Z + 2$$

b. By dividing to triangles $E S Z$, $Q W A$, $W A S$, $W E S$, and $A S Z$. The four latter triangles all have base $1$, height $1$, and area $1/2$, so
$$\text{Area } Q A Z E = \text{Area } E S Z + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \operatorname{Area} E S Z + 2$$

Since quadrilateral $Q A Z E$ must have the same area in both computations, we deduce that $\text{Area } Q A Z = \text{Area } E S Z$.