Learning about angles in circles is fundamental. This concept was extensively addressed by Euclid and the Greeks in general, such as Pythagoras and Archimedes.

Bow

Definition: Let $A$ and $B$ be points on a circle in a plane. The straight line $\vec{AB}$ (arc) separates the plane into two half-planes, each containing a part of the circle.

Central Angle

Definition: Let $A$ and $B$ be points of a circle on a center plane $C$. The central angle $\angle ACB$ determined by $A$ and $B$ or determined by the chord $AB$.

Inscribed Angle

Definition: Let $A$ and $B$ be points of a circle on a center plane $C$. The inscribed angle $\angle ACB$ is determined when all vertices are on the circle.

Inscribed angle equivalent to half of an arc

Theorem 1: Every angle inscribed in a circle has half the measure of the corresponding arc.

Angle inscribed in a semicircle equivalent to a right angle

Lemma 1: In a circle, if an inscribed angle projects a semicircle, its measure is $90^\circ.$

Projected inscribed angle

Corollary 1: In a circle, if two inscribed angles project the same arc, then they have the same measure.

These results show that the rule of angles opposite the vertex described in the post angle property applies.

That’s it for now. Wait for the next posts.