Question

which statements must be true about the inscribed quadrilateral?
choose all that apply.
a. ∠j is a right angle because (overparen{kgh}) is a semicircle.
b. ∠g ≅ ∠j because they are inscribed angles that intercept arcs of the same measure.
c. ∠h ≅ ∠k because they are inscribed angles that intercept arcs of the same measure.
d. ∠g is supplementary to ∠j because they are inscribed angles intercepting arcs that add to (360^{circ}).
e. ∠g is supplementary to ∠k because they are inscribed angles intercepting arcs that add to (360^{circ}).

Explanation:

Step1: Recall properties of inscribed quadrilaterals

In a cyclic quadrilateral (inscribed quadrilateral in a circle), the sum of opposite angles is \(180^{\circ}\) (they are supplementary). Also, an inscribed angle is half the measure of its intercepted arc.

Step2: Analyze each option

• Option A: For \(\angle J\) to be a right - angle, the arc \(KGH\) must be a semicircle (\(180^{\circ}\)). But there is no information given that \(KGH\) is a semicircle. So, this option is not necessarily true.
• Option B: \(\angle G\) and \(\angle J\) do not intercept arcs of the same measure. The arcs intercepted by \(\angle G\) and \(\angle J\) are different. So, this option is not true.
• Option C: \(\angle H\) and \(\angle K\) do not intercept arcs of the same measure. The arcs intercepted by \(\angle H\) and \(\angle K\) are different. So, this option is not true.
• Option D: Let the arc intercepted by \(\angle G\) be \(x\) and the arc intercepted by \(\angle J\) be \(y\). We know that \(x + y=360^{\circ}\) (the sum of arcs in a circle). Since \(\angle G=\frac{1}{2}x\) and \(\angle J = \frac{1}{2}y\), then \(\angle G+\angle J=\frac{1}{2}(x + y)=\frac{1}{2}\times360^{\circ}=180^{\circ}\). So, \(\angle G\) is supplementary to \(\angle J\).
• Option E: Let the arc intercepted by \(\angle G\) be \(m\) and the arc intercepted by \(\angle K\) be \(n\). We know that \(m + n

eq360^{\circ}\). In a cyclic quadrilateral, \(\angle G+\angle K = 180^{\circ}\) because \(\angle G\) and \(\angle K\) are opposite angles. But the reason given in the option is wrong. The correct reason for \(\angle G\) and \(\angle K\) being supplementary is that they are opposite angles of a cyclic quadrilateral.

Answer:

D. \(\angle G\) is supplementary to \(\angle J\) because they are inscribed angles intercepting arcs that add to \(360^{\circ}\)