Question

this quadrilateral is a kite. ∠q and ∠s are vertex angles. find ( mangle qrt ).

Explanation:

Step1: Recall kite vertex angle property

In a kite, the vertex angles are bisected by the diagonal connecting them, so $\angle PQR$ is split into two equal angles: $\angle PQT = \angle QRT$. Also, $\angle QPT = 90^\circ$ (diagonals of a kite are perpendicular at intersection).

Step2: Calculate $\angle PQT$

In right triangle $PQT$, sum of angles is $180^\circ$.
$\angle PQT = 180^\circ - 90^\circ - 37^\circ = 53^\circ$

Step3: Relate to $\angle QRT$

Since diagonal $QS$ bisects vertex $\angle PQR$, $\angle QRT = \angle PQT$.

Answer:

$53^\circ$