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Central Angles and Intercepted Arcs - Concept
A central angle is an angle whose vertex is on the center of the circle and whose endpoints are on the circle. The endpoints on the circle are also the endpoints for the angle's intercepted arc. The angle measure of the central angle is congruent to the measure of the intercepted arc which is an important fact when finding missing arcs or central angles.
In the circle universe there are two related and key terms, there are central angles and intercepted arcs.
We'll start off with central angle, key facet of a central angle is that its the vertex is that the center of the circle. This equation down here says that the measure of angle abc which is our central angle is equal to the measure of the arc ac. Well we call that arc ac the intercepted arc just like a football pass intercept, so from a to c notice those are also the place where the central angle intersects the circle so this is called our intercepted arc and for central angles they will always be congruent to their intercepted arc and this picture right here I've drawn something that is not a central angle. The reason is its vertex is on the circle not at the center of the circle.
If we look at congruent chords in a circle so I've drawn 2 congruent chords I've said 2 important things that congruent chords have congruent central angles which means I can say that these two central angles must be congruent and how could I prove that? Well if you look at these two sides that I have marked congruent and if you look at the other two sides of the triangle we see that they are radii so these two are congruent and these 2 radii are all congruent so we could use the side side side conjecture to say that these two triangles must be congruent therefore their central angles are also congruent.
One other consequence of this is that they also will have congruent intercepted arcs so I could say that this arc right here which is formed by that congruent chord is congruent to that intercepted arc so lots of interesting things going over central angles and intercepted arcs that'll help us find missing measures.