(a) A pentagon is such that one of its exterior sides is 60°. Two others are (90 - m)° each while the remaining angles are (30 + 2m)° each. Find the value o...

Answer Details

(a) The exterior angles of any polygon add up to \(360^\circ\). The pentagon has five exterior angles: one is \(60^\circ\), two are \((90-m)^\circ\) each, and the remaining two are \((30+2m)^\circ\) each. Therefore:

\[60+2(90-m)+2(30+2m)=360\]\[60+180-2m+60+4m=360\]\[300+2m=360\]\[2m=60\Rightarrow m=30\]

(b) From the diagram PQR is a straight line, \(\overline{QR}=\sqrt{3}\text{ cm}\), \(\overline{SQ}=2\text{ cm}\), and the right-angle mark at R shows \(SR\perp QR\). So triangle SQR is right-angled at R, with SQ the hypotenuse.

In right-angled triangle SQR, using \(\angle SQR\):

\[\cos(\angle SQR)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{\overline{QR}}{\overline{SQ}}=\frac{\sqrt{3}}{2}\]\[\angle SQR=\cos^{-1}\!\left(\tfrac{\sqrt{3}}{2}\right)=30^\circ\]

Since PQR is a straight line, \(\angle PQS\) and \(\angle SQR\) are angles on a straight line at Q:

\[\angle PQS=180^\circ-\angle SQR=180^\circ-30^\circ=150.0^\circ\]