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

Answer 1

Let AB be a diameter of a circle with centre O, and let C be any other point on the circle. The inscribed angle theorem says an inscribed angle equals half the measure of its intercepted arc (equivalently, half the measure of the central angle that subtends the same arc).

The arc AB determined by a diameter is a semicircle, so its measure is 180°. The central angle AOB that subtends arc AB therefore measures 180°. By the inscribed angle theorem the inscribed angle ACB that subtends the same arc measures half of that:
∠ACB = 1/2 · ∠AOB = 1/2 · 180° = 90°.

Thus any angle inscribed in a semicircle is a right angle. (The converse is also true: a triangle is right-angled exactly when its hypotenuse is a diameter — Thales' theorem.)