We are given a kite $WXYZ$. We know that $m\angle Y = 40^\circ$ and $m\angle W = 112^\circ$. We need to find $m\angle X$.

1. Problem Description

We are given a kite $WXYZ$. We know that $m\angle Y = 40^\circ$ and $m\angle W = 112^\circ$. We need to find $m\angle X$.

2. Solution Steps

A kite is a quadrilateral, and the sum of the interior angles of any quadrilateral is $360^\circ$.

$$m\angle W + m\angle X + m\angle Y + m\angle Z = 360^\circ$$

In a kite, the angles between the non-congruent sides are congruent.

Thus, in kite $WXYZ$, we have $\angle X \cong \angle Z$, which means $m\angle X = m\angle Z$.

We are given $m\angle Y = 40^\circ$ and $m\angle W = 112^\circ$. Substituting these values into the equation for the sum of the interior angles, we get

$$112^\circ + m\angle X + 40^\circ + m\angle Z = 360^\circ$$

Since $m\angle X = m\angle Z$, we can write

$$112^\circ + m\angle X + 40^\circ + m\angle X = 360^\circ$$

$$152^\circ + 2(m\angle X) = 360^\circ$$

$$2(m\angle X) = 360^\circ - 152^\circ$$

$$2(m\angle X) = 208^\circ$$

$$m\angle X = \frac{208^\circ}{2}$$

$$m\angle X = 104^\circ$$

3. Final Answer

$m\angle X = 104^\circ$