Question

If m∠A = 43° and the measure of arc AB = 69°, then m∠B = ° (Blank 1). (Round your answer to one decimal place, if necessary.)

Answer 1

Answer:  102.5 degrees

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Explanation:

If angle A is 43 degrees, then minor arc BC is 2*43 = 86 degrees according to the inscribed angle theorem. The central angle is twice that of the inscribed angle. Both of these angles subtend the same minor arc.

When I say "minor arc BC", I mean that we go from B to C along the shortest path. Any minor arc is always less than 180 degrees.

Since minor arc AB is 69 degrees, and minor arc BC is 86 degrees, this means arc ABC is arcAB+arcBC = 69+86 = 155 degrees

Let's say point D is some point on the circle that isn't between A and B, and it's not between B and C either. Refer to the diagram below. The diagram is to scale. The diagram your teacher provided is not to scale because arc ABC is way too big (it appears to be over 180 degrees). Hopefully the diagram below gives you a better sense of what's going on.

Because arc ABC = 155 degrees, this means the remaining part of the circle, arc ADC, is 360-(arc ABC) = 360-155 = 205 degrees

Inscribed angle B subtends arc ADC. So we'll use the inscribed angle theorem again, but this time go in reverse from before. We'll cut that 205 degree angle in half to get 205/2 = 102.5 degrees which is the measure of angle B. This value is exact. In this case, we don't need to apply any rounding.