Question

given qrst is inscribed in the circle, prove opposite angles in a quadrilateral are supplementary.
move angle types to the blanks and move expressions to the boxes to complete the proof.
in qrst, let the measure of \\( \angle stq = x^{\circ} \\).
what type of angle is \\( \angle stq \\)? __________. this means \\( m\widehat{qrs}=\\)
because the arcs of a circle add up to \\( 360^{\circ}, m\widehat{stq}=\\)
what type of angle is \\( \angle qrs \\)? __________. this means \\( m\angle qrs = \\)
therefore, opposite angles of a quadrilateral inscribed in a circle are always supplementary.
a central angle an inscribed angle a straight angle
x 2x \\( (180 - x) \\)
\\( (180 + x) \\) \\( (360 - x) \\) \\( (360 - 2x) \\)

Explanation:

Step1: Determine the type of \(\angle STQ\) and find \(m\overarc{QRS}\)

\(\angle STQ\) is an inscribed angle. By the inscribed - angle theorem, if an inscribed angle has a measure of \(x^{\circ}\), then the measure of the arc it intercepts is \(2x^{\circ}\). So \(m\overarc{QRS}=2x\).

Step2: Find \(m\overarc{STQ}\)

Since the sum of the measures of the arcs of a circle is \(360^{\circ}\), and \(m\overarc{QRS} = 2x\), then \(m\overarc{STQ}=360 - 2x\).

Step3: Determine the type of \(\angle QRS\) and find \(m\angle QRS\)

\(\angle QRS\) is an inscribed angle. The arc \(\overarc{STQ}\) is intercepted by \(\angle QRS\). Using the inscribed - angle theorem \(m\angle QRS=\frac{1}{2}m\overarc{STQ}\). Substitute \(m\overarc{STQ}=360 - 2x\) into the formula: \(m\angle QRS=\frac{1}{2}(360 - 2x)=180 - x\).

Answer:

\(\angle STQ\): An inscribed angle, \(m\overarc{QRS}=2x\), \(m\overarc{STQ}=360 - 2x\), \(\angle QRS\): An inscribed angle, \(m\angle QRS = 180 - x\)