Inscribed Angles - Concept


Inscribed angles are angles whose vertices are on a circle and that intersect an arc on the circle. The measure of an inscribed angle is half of the measure of the intercepted arc and half the measure of the central angle intersecting the same arc. Inscribed angles that intercept the same arc are congruent.


Inscribed angles are different from central angles because their vertex is on this is on the circle so if I were to draw in two radii which would form a central angle aoc there's a special relationship between the central angle and this inscribed angle when they share the same intercepted arc from a to c and that special relationship is written in these two equations. First one says that this inscribed angle abc is equal to half of this central angle aoc. Another way of saying that is in terms of the intercepted arc. If we have an intercepted arc ac its measure is going to be twice as much as the intercepted arc. If I had to flip that around, this is going to be half of the intercepted arc.
The second key thing about inscribed angles are when they intercept the same arc so here we have one inscribed angle its vertex is on the circle and I have two endpoints, one right here and one right here, and if I were to pick a random point somewhere on this circle and draw in two chords to where the endpoints of the other inscribed angle are so now we've created two inscribed angles that have the same intercepted arc. If they have the same intercepted arc then they must be congruent, so that's going to be helpful when you're trying to problem solve when you have inscribed angles and remember the other key thing is that an inscribed angle is equal to half of it's central angle and half of the intercepted arc.

inscribed angle intercepted arc central angle congruent arcs