Asked by john
Circle F is inscribed with quadrilateral E D C B. Point A is on the circle between points E and B. Angle D E B is 70 degrees and angle D B C is 80 degrees. The measure of arc D C is 88 degrees.
What is the measure of Arc E A B in circle F?
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Answered by
GPT-5 mini
AI
Answer: 60°
Reason: Inscribed-angle theorem. ∠DEB = 70° intercepts arc DB, so arc DB = 2·70 = 140°. (The 80° angle must be ∠DBE; it intercepts arc DE, so arc DE = 2·80 = 160°.) Given arc DC = 88°, arc CB = arc DB − arc DC = 140° − 88° = 52°. The four arcs sum to 360°, so arc EB = 360° − (arc DC + arc CB + arc DE) = 360° − (88°+52°+160°) = 60°. Thus arc EAB = 60°.
Reason: Inscribed-angle theorem. ∠DEB = 70° intercepts arc DB, so arc DB = 2·70 = 140°. (The 80° angle must be ∠DBE; it intercepts arc DE, so arc DE = 2·80 = 160°.) Given arc DC = 88°, arc CB = arc DB − arc DC = 140° − 88° = 52°. The four arcs sum to 360°, so arc EB = 360° − (arc DC + arc CB + arc DE) = 360° − (88°+52°+160°) = 60°. Thus arc EAB = 60°.
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