3. Given PQ=24,PS=19,PR=42,TQ=10,mangle PQR=106^circ ,mangle QSR=49^circ and mangle PRS=35^circ QR=19^circ mangle QRS=35^circ SR=24^circ mangle PQS=49^circ PT=21^circ mangle RPS=39^circ SQ=20^circ mangle PSQ=

The question seems to be asking for the measurement of certain angles within the quadrilateral PQRS, given various side lengths and angles. However, there seems to be a confusion in the way the question is presented. The options provided (e.g., "QR=19°") are not consistent with the context of the question, as they seem to mix up side lengths with angle measurements (degrees are a unit of angle, not length). Let's clarify the given information and then proceed to find the missing angle measurements:Given:- PQ = 24- PS = 19- PR = 42- TQ = 10- ∠PQR = 106°- ∠QSR = 49°- ∠PRS = 35°We are asked to find:- ∠PQS- ∠RPSSince ∠PQS = ∠RPS (as given in the picture details), we only need to find one of these angles.To find ∠PQS or ∠RPS, we can use the fact that the sum of the angles in a quadrilateral is 360°. We can write an equation that sums all the angles in quadrilateral PQRS and solve for the unknown angle:∠PQS + ∠PQR + ∠PRS + ∠QSR = 360°We know that ∠PQS = ∠RPS, and we have the measurements for the other angles, so we can substitute the known values into the equation:2∠PQS + 106° + 35° + 49° = 360°Now, let's solve for ∠PQS:2∠PQS + 190° = 360°2∠PQS = 360° - 190°2∠PQS = 170°∠PQS = 170° / 2∠PQS = 85°Therefore, the measure of angle ∠PQS (and also ∠RPS, since they are equal) is 85°.