In figure 3.57, □PQRS is cyclic. side PQ ≅ side RQ. \(\angle PSR = 110^\circ\), Find-
(1) measure of \(\angle PQR\) (2) m(arc PQR) (3) m(arc QR) (4) measure of \(\angle PRQ\)
Figure: Diagonal PR is drawn.
Answer: (1) \(70^\circ\) (2) \(220^\circ\) (3) \(110^\circ\) (4) \(55^\circ\)
(1) Opposite angles of a cyclic quadrilateral are supplementary:
\[\angle PQR = 180^\circ - 110^\circ = 70^\circ\](2) \(\angle PSR\) is inscribed in arc PSR and intercepts arc PQR:
\[m(\text{arc PQR}) = 2 \times 110^\circ = 220^\circ\](3) Chord PQ ≅ chord RQ, so arc PQ ≅ arc QR. Together they form arc PQR:
\[m(\text{arc QR}) = \frac{220^\circ}{2} = 110^\circ\](4) \(\angle PRQ\) intercepts arc PQ (\(110^\circ\)):
\[\angle PRQ = \frac{1}{2} \times 110^\circ = 55^\circ\]Check: in \(\triangle PQR\), \(70^\circ + 55^\circ + 55^\circ = 180^\circ\).
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