Asked by Ian
Quadrilateral ABCD is inscribed in a circle. The m angle B=91, m BC =54, and m CD=106. Find the missing angle measures. m angle A=_ degrees, m angle C=_ degrees, and m angle D=_ degrees.
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GPT-5 mini
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Assume the vertices go around the circle in order A–B–C–D. Use that an inscribed angle = half the measure of its intercepted arc and opposite angles of a cyclic quadrilateral are supplementary.
1. Angle B = 91° ⇒ intercepted arc AC = 2·91 = 182°. That arc AC = arc AD + arc DC, so arc AD = 182 − arc CD = 182 − 106 = 76°.
2. arc BD (the short way, through C) = arc BC + arc CD = 54 + 106 = 160°. Angle A intercepts that arc, so angle A = 1/2·160 = 80°.
3. Opposite angles are supplementary: angle C = 180 − angle A = 180 − 80 = 100°.
4. Also angle D = 180 − angle B = 180 − 91 = 89°.
Answers: m∠A = 80°, m∠C = 100°, m∠D = 89°.
1. Angle B = 91° ⇒ intercepted arc AC = 2·91 = 182°. That arc AC = arc AD + arc DC, so arc AD = 182 − arc CD = 182 − 106 = 76°.
2. arc BD (the short way, through C) = arc BC + arc CD = 54 + 106 = 160°. Angle A intercepts that arc, so angle A = 1/2·160 = 80°.
3. Opposite angles are supplementary: angle C = 180 − angle A = 180 − 80 = 100°.
4. Also angle D = 180 − angle B = 180 − 91 = 89°.
Answers: m∠A = 80°, m∠C = 100°, m∠D = 89°.
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