Inscribed angles where one chord is a diameter

The measure of an inscribed angle is equal to half of the measure of the arc between its sides. Considering that the arc of a semicircle is 180º, any angle inscribed in a semicircle has half that value, that is 90º.

Any angle inscribed in a semicircle is right.

Created with GeoGebra by Vitor Nunes

Interactivity

You can move point A along the entire semicircle. Thus you will verify that whatever his position, he measures always 90º.

Also verify that the sum of the 3 internal angles of the triangle is always 180º.

Applications

The inscribed angle theorem is used in many proofs of elementary Euclidean geometry of the plane. A special case of the theorem is Thales' theorem, which states that the angle subtended by a diameter is always 90°, i.e., a right angle. As a consequence of the theorem, opposite angles of cyclic quadrilaterals sum to 180°; conversely, any quadrilateral for which this is true can be inscribed in a circle.