Question

in the figure below, triangle qrt, triangle rsq, and triangle qst are scalene triangles.
if m∠qrs = 37° and m∠tqs = 34°, what is m∠tqr?
a. 109°
b. 87°
c. 90°
d. 93°

Explanation:

Step1: Recall angle - sum property of a right - triangle

In right - triangle QST, $\angle QST = 90^{\circ}$. In $\triangle QRS$, we know $\angle QRS=37^{\circ}$. Using the angle - sum property of a triangle ($\angle QRS+\angle RQS+\angle QSR = 180^{\circ}$) in $\triangle QRS$ and since $\angle QSR = 90^{\circ}$, we can find $\angle RQS$.
$\angle RQS=180^{\circ}-\angle QRS - \angle QSR=180^{\circ}-37^{\circ}-90^{\circ}=53^{\circ}$.

Step2: Find $\angle TQR$

We know that $\angle TQR=\angle TQS+\angle SQR$. Given $\angle TQS = 34^{\circ}$ and we found $\angle SQR = 53^{\circ}$.
$\angle TQR=34^{\circ}+53^{\circ}=87^{\circ}$.

Answer:

B. $87^{\circ}$