Answer:
a) 140°
b) 20°
c) 27°
Explanation:
You want the measures of various arcs and angles in the given figure.
Inscribed angle
An angle inscribed in a circle has half the measure of the arc it subtends. The angle at a tangent (angle NRQ, for example) is a special case. It, too, has half the measure of the arc subtended.
a) Arc NPR
Arc NPR is the arc subtended by the angle NRQ, so will have double the measure of that angle.
arc NPR = 2 × ∠NRQ = 2×70°
arc NPR = 140°
b) Angle RNS
Secant NS divides the circle into two arcs of 180° each. As we found in part (a), arc NR is 140°, so the remaining arc in that half of the circle is ...
180° -140° = 40°
Inscribed angle RNS has half the measure of this arc, so is ...
∠RNS = (1/2)(40°)
∠RNS = 20°
c) Angle RNQ
Angle RNQ subtends arc PR, which is what is left of 140° arc NR after 86° arc NP is subtracted.
arc PR = 140° -86° = 54°
Angle RNQ is half the measure of this arc, so is ...
∠RNQ = (1/2)(54°)
∠RNQ = 27°
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