I'm assuming you want to answer problem 5.
If so, then the answer is A = 84 degrees
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Step-by-step explanation:
Minor arc BC is 96 degrees. This is the measure along the edge of the curve from B to C, following the shorter path. You correctly identified that the central angle BKC is also congruent to this arc measure, since the central angle cuts off this arc.
Because segments AB and AC are tangents to the circle, this makes the segments to be perpendicular to the radii that meet at the point of tangency. In other words, AB is perpendicular to BK, and AC is perpendicular to CK.
Ignore line BC. Focus on quadrilateral ABKC. So far we know that
- angle K = 96
- angle B = 90
- angle C = 90
The only thing missing is angle A, which is what we're after. Recall that for any quadrilateral, the four interior angles always add to 360 degrees.
So,
A+B+K+C = 360
A+90+96+90 = 360
A+276 = 360
A = 360-276
A = 84
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Or as a shortcut, you can use the property that because B and C are right angles, this means we have 360-B-C = 360-90-90 = 180 degrees left for angles A and K to add to. This makes angle A and angle K to be supplementary.
So,
A+K = 180
A+96 = 180
A = 180-96
A = 84