Circle Angle Relationships - Math 3

In circle R, inscribed angle DEF intercepts arc DF that measures $128°$ . Point G lies on arc DF such that arc DG = $48°$ . What is the measure of angle DEG?
Answer: $24°$ Inscribed angle DEF = 128°/2 = 64°. Since G lies on arc DF with arc DG = 48°, the inscribed angle DEG subtends arc DG. Therefore, angle DEG = 48°/2 = 24°. Choice B would be 80°/2 if arc DG were 80°. Choice C gives the arc measure rather than the inscribed angle. Choice D gives the full angle DEF.
In circle V, diameter AB passes through the center. Point C is on the circle such that arc AC = $116°$ . A tangent at point C intersects the extension of diameter AB at point D. What is the measure of angle ACD?
Answer: $32°$ Since AB is a diameter, arc AB = 180°. Given arc AC = 116°, then arc CB = 180° - 116° = 64°. The angle between a tangent and a chord equals half the arc cut off by the chord on the side away from the tangent. Since the tangent at C and chord CA form angle ACD, this angle equals half of arc CB (the arc on the opposite side of chord AC from the tangent). Therefore, angle ACD = (arc CB)/2 = 64°/2 = 32°.