(a) A pentagon is such that one of its exterior sides is 60^\circ. Two others are (90 - m)^\circ each while the remaining angles are (30 + 2m)^\circ each. Find the value of m.
(b)
In the diagram, PQR is a straight line, \(\overline{QR} = \sqrt{3} cm\) and \(\overline{SQ} = 2 cm\). Calculate, correct to one decimal place, < PQS.
Verified Correct Answer: .
Step-by-Step Mathematical Solution
The sum of exterior angles = 360^\circ
\(60^\circ + 2(90 - m)^\circ + 2(30 + 2m)^\circ = 360^\circ\)
\(60^\circ + 180^\circ - 2m^\circ + 60^\circ + 4m^\circ = 360^\circ\)
\(300^\circ + 2m = 360^\circ\)
\(\implies 2m = 360^\circ - 300^\circ = 60^\circ\)
\(m = 30^\circ\)
(b)
In \(\Delta QRS, \cos Q = \frac{\sqrt{3}}{2}\)
\(\cos Q = 0.8660 \implies Q = 30^\circ\)
\(\therefore < SQR = 30^\circ^\)
\(< PQS = 180^\circ - 30^\circ \) (angles on a straight line = 180^\circ)
\(< PQS = 150^\circ\)
Source: Myschool Past Questions Archive (Record #52938)