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Central Angles and Intercepted Arcs - Problem 1
A central angle is an angle with its vertex at the center of a circle, whose legs intersect the circle. The measure of a central angle is the same as the measure of the arc of its endpoints. Therefore, if central angle ∠ABC has measure 155°, then arc AC also has a measure of 155°.
Recall that a chord is a line that connects the intersections of a central angle with the circle. If two chords are congruent, their central angles are also congruent.
We can apply what we know about central angles and congruent chords in a circle to a problem like this. Well, we’re trying to find two different things, one the measure of angle B and two the measure of arc A.
Let’s start with the measure of arc A. if I look at A, it is the intercepted arc of this central angle, 155 degrees. So if I go back to what I know about a central and its intercepted arc, we said that they’re going to be equal to each other, which means A has to be equal to 155 degrees. Also this is the measure of arc A, is 155 degrees.
Second variable we’re solving for is B. If I look at B, it is the central angle that intersects a chord that is congruent to this central angle of 82 degrees, which means these two must be congruent something that we noticed when we said congruent chords have congruent central angles and congruent arcs, which means if B is congruent to 82, then B must be 82 degrees.
So the key here was remembering that chords have congruent, if the chords are congruent they have congruent central angle and that the central angle is congruent to its intercepted arc.