Answer
$y = 57^{\circ}$
$x = 57^{\circ}$
$z = 66^{\circ}$
Work Step by Step
We can see that this quadrilateral is a rhombus because all sides are congruent.
We can also see the diagonal is bisecting one of the angles. If one angle formed by the bisection is $57^{\circ}$, then the other angle resulting from the bisection, $\angle y$, is also $57^{\circ}$.
The two triangles formed by the diagonal of the rhombus are isosceles because two sides are marked congruent. The angles opposite to these sides are also congruent; therefore, $\angle y ≅ \angle x$, so $m \angle x = 57^{\circ}$.
Now, we can use the triangle sum theorem to determine the third angle, $\angle z$:
The triangle sum theorem states that the sum of the measures of the interior angles of a triangle equals $180^{\circ}$:
$m \angle z = 180 - (m \angle x + m \angle y)$
Let's plug in the measures for $\angle x$ and $\angle y$:
$m \angle z = 180 - (57 + 57)$
Evaluate what is in parentheses first:
$m \angle z = 180 - (114)$
Subtract to solve:
$m \angle z = 66^{\circ}$
