Question

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:

1. Recall the inscribed angle theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
2. For an angle inscribed in a semicircle, the intercepted arc is a semicircle. A semicircle measures \( 180^\circ \) (since a full circle is \( 360^\circ \), a semicircle is \( \frac{360^\circ}{2}=180^\circ \)).
3. Let the inscribed angle be \( \theta \) and its intercepted arc be \( 180^\circ \). By the inscribed angle theorem, \( \theta=\frac{1}{2}\times \text{measure of intercepted arc} \).
4. Substitute the measure of the intercepted arc: \( \theta = \frac{1}{2}\times180^\circ=90^\circ \). An angle with measure \( 90^\circ \) is a right angle.

Answer:

To justify the corollary that an angle inscribed in a semicircle is a right angle using the inscribed angle theorem:

1. Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
2. Intercepted Arc for Semicircle: An angle inscribed in a semicircle intercepts a semicircular arc. A full circle has \( 360^\circ \), so a semicircle has \( \frac{360^\circ}{2}=180^\circ \).
3. Apply the Theorem: Let the inscribed angle be \( \theta \). By the theorem, \( \theta=\frac{1}{2}\times \text{measure of intercepted arc} \). Substituting the semicircle’s arc measure (\( 180^\circ \)): \( \theta=\frac{1}{2}\times180^\circ = 90^\circ \).
4. Conclusion: A \( 90^\circ \) angle is a right angle, so an angle inscribed in a semicircle is a right angle.