Question

justifying the second corollary to the inscribed angle theorem. 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.

Explanation:

Step1: Recall inscribed - angle theorem

The inscribed - angle theorem states that the measure of an inscribed angle $\theta$ in a circle is half the measure of the central angle $\alpha$ that subtends the same arc, i.e., $\theta=\frac{1}{2}\alpha$.

Step2: Consider a semi - circle

In a semi - circle, the central angle corresponding to the arc of the semi - circle is $180^{\circ}$. Let the inscribed angle be $\angle ABC$ and the central angle be $\angle AOC$ (where $O$ is the center of the circle).

Step3: Apply the inscribed - angle theorem

If $\alpha = 180^{\circ}$ (central angle of a semi - circle), then by the inscribed - angle theorem $\theta=\frac{1}{2}\alpha$. Substituting $\alpha = 180^{\circ}$ into the formula, we get $\theta=\frac{1}{2}\times180^{\circ}=90^{\circ}$. So an angle inscribed in a semi - circle is a right angle.

Answer:

An angle inscribed in a semi - circle is a right angle because by the inscribed - angle theorem, which states that the measure of an inscribed angle is half the measure of the central angle subtending the same arc, and since the central angle of a semi - circle is $180^{\circ}$, the inscribed angle is $\frac{1}{2}\times180^{\circ}=90^{\circ}$.