Question

margaret measures \\( \angle h g k \\) with a protractor and says that it is \\( 98 ^ { \circ } \\). is margarets answer reasonable? explain. margarets answer \\( \square \\) reasonable. \\( \overparen { g j k } \\) is \\( \square \\) than a semicircle, so its measure is \\( \square \\) than \\( \square \\). because \\( \angle h g k \\) intercepts \\( \overparen { g j k } \\), \\( m \angle h g k \\) is \\( \square \\) than \\( \square \\).

Explanation:

Step1: Recall the property of an inscribed angle and its intercepted arc

The measure of an inscribed angle is half the measure of its intercepted arc. A semicircle has a measure of \(180^{\circ}\).

Step2: Analyze the arc \(\overset{\frown}{GJK}\)

Since \(\overset{\frown}{GJK}\) is larger than a semicircle (a major arc), its measure \(m\overset{\frown}{GJK}> 180^{\circ}\)

Step3: Calculate the measure of \(\angle HGK\)

We know that \(m\angle HGK=\frac{1}{2}m\overset{\frown}{GJK}\). If \(m\overset{\frown}{GJK}>180^{\circ}\), then \(m\angle HGK=\frac{1}{2}m\overset{\frown}{GJK}>90^{\circ}\)

Answer:

Margaret's answer is reasonable. \(\overset{\frown}{GJK}\) is larger than a semicircle, so its measure is greater than \(180^{\circ}\). Because \(\angle HGK\) intercepts \(\overset{\frown}{GJK}\), \(m\angle HGK\) is greater than \(90^{\circ}\)