Question

a student wants to explain how you can use the inscribed angle theorem to justify its second corollary, that an angle inscribed in a semicircle is a right angle.

the student says that a circle measures 360 degrees, so a semicircle measures 180 degrees. by using the inscribed angle theorem, the measure of the inscribed angle would be half of 180 degrees, or 90 degrees, which is a right angle.

determine if the student’s argument is correct or incorrect and justify your decision. for any portion of the argument in which there is an error, provide and explain an alternative argument. if you wish to include an image in your explanation, you can draw the image on paper, take a picture, and insert it into your response using the image tool.

Explanation:

The student's argument is correct. The inscribed - angle theorem states that the measure of an inscribed angle in a circle is half the measure of its intercepted arc. A semicircle is an arc that measures \(180^{\circ}\). When an angle is inscribed in a semicircle, the intercepted arc is the semicircle itself (\(180^{\circ}\)). By the inscribed - angle theorem, the measure of the inscribed angle \(\theta=\frac{1}{2}\times\) (measure of the intercepted arc). Substituting the measure of the semicircle (\(180^{\circ}\)) into the formula, we get \(\theta = \frac{1}{2}\times180^{\circ}=90^{\circ}\), which is a right angle.

Answer:

The student's argument is correct.