Question

quadrilateral pqrs is inscribed in the circle shown, and pqr is a semi - circle. move the options to the blanks to correctly complete the sentences. the angles must each measure _ because the measure of an inscribed angle is _ the measure of its intercepted arc. ∠p and ∠r must each measure 90° because the measure of an inscribed angle is half the measure of its intercepted arc. ∠q and ∠s must each measure 180° because the measure of an inscribed angle is half the measure of its intercepted arc.

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

Step2: Analyze \(\angle P\) and \(\angle R\)

Since \(\widehat{PQR}\) is a semi - circle and \(\angle P\) and \(\angle R\) are inscribed angles that intercept semi - circles. The measure of a semi - circle is \(180^{\circ}\). By the inscribed - angle theorem, the measure of \(\angle P\) and \(\angle R\) is \(\frac{1}{2}\times180^{\circ}=90^{\circ}\).

Step3: Analyze \(\angle Q\) and \(\angle S\)

Opposite angles of a cyclic quadrilateral (a quadrilateral inscribed in a circle) are supplementary. So \(\angle Q+\angle S = 180^{\circ}\).

Answer:

\(\angle P\) and \(\angle R\) must each measure \(90^{\circ}\) because the measure of an inscribed angle is half the measure of its intercepted arc. \(\angle Q\) and \(\angle S\) are supplementary (sum to \(180^{\circ}\)) because \(PQRS\) is a cyclic quadrilateral.