In the diagram, PT is parallel to QR. What is the measure of angle PQR in degrees?

116

Step-by-step explanation:

Since the two angles at $R$ add to $180^\circ,$ then $\angle QRT + 128^\circ = 180^\circ,$ so $\angle QRT = 52^\circ.$

Since $PT$ and $QR$ are parallel, then $\angle PTR$ and $\angle QRT$ are supplementary, so $2x^\circ + 52^\circ = 180^\circ$ or $2x^\circ = 128^\circ$ or $x = 64.$

Therefore, three of the angles of quadrilateral $PQRT$ are $64^\circ,$ $128^\circ,$ and $52^\circ.$

Since the angles in a quadrilateral add to $360^\circ,$ then $\angle PQR = 360^\circ - 64^\circ - 128^\circ - 52^\circ = \boxed{116} \text{ degrees}$.

2. angles qrt and qst both measures 156 degrees.

step-by-step explanation:  

we have been given a diagram of a circle and we are told that measure of arc qvt is 156 degrees.  

we can see from our given diagram that angle qrt and angle qst subtends to arc qvt.

since we know that angles subtended by the same arc at the circumference are always equal.

by angle subtended by same arc theorem, measure of angle qrt and angle qst will be equal to the measure of arc qvt.

therefore, angles qrt and qst both measures 156 degrees and 2nd statement is the correct choice.