Answer
The measurement of angle R is \[70{}^\circ \].
Work Step by Step
In \[\Delta PTQ\], \[\angle PTQ\]is \[70{}^\circ \] and \[\angle TPQ\] is \[60{}^\circ \], and in the \[\Delta SQR\], \[\angle QSR\] is \[30{}^\circ \]and also, \[\angle TQS\] is \[50{}^\circ \].
According to angle sum property, the sum of all the three angles of a triangle is \[{{180}^{\circ }}\]. Compute the measure of angle PQT as follows:
In \[\Delta PTQ\] by angle sum property the sum of all the three angels of a triangle is \[{{180}^{\circ }}\]. Therefore, \[\angle PTQ+\angle TPQ+\angle PQT=180{}^\circ \]
Compute the measurement of angle PQT as shown below:
\[\begin{align}
& 70{}^\circ +60{}^\circ +\angle PQT=180{}^\circ \\
& 130{}^\circ +\angle PQT=180{}^\circ \\
& \angle PQT=180{}^\circ -130{}^\circ \\
& \angle PQT=50{}^\circ
\end{align}\]
Compute the measurement of angle SQR using the fact that angles PQT, TQS, and SQR form a straight angle. Therefore, their sum is equal to\[{{180}^{\circ }}\].
\[\begin{align}
& \angle PQT+\angle TQS+\angle SQR=180{}^\circ \\
& 50{}^\circ +50{}^\circ +\angle SQR=180{}^\circ \\
& \angle SQR=180{}^\circ -100{}^\circ \\
& \angle SQR=80{}^\circ
\end{align}\]
In \[\Delta SQR\] by angle sum property the sum of all the three angels of a triangle is \[{{180}^{\circ }}\]. Therefore, \[\angle SQR+\angle QSR+\angle SRQ=180{}^\circ \]
\[\begin{align}
& 80{}^\circ +30{}^\circ +\angle SRQ=180{}^\circ \\
& 110{}^\circ +\angle SRQ=180{}^\circ \\
& \angle SRQ=180{}^\circ -110{}^\circ \\
& \angle SRQ=70{}^\circ
\end{align}\]
Hence, the measurement of angle R is \[70{}^\circ \].
