Given: circle k(O), mAM =125°, mEF =31°, m∠MAF=75° Find: m∠AME

Answer:[tex]m<A.M.E=58\°[/tex]Step-by-step explanation:step 1Find the measure of angle M.E.Fwe know that In an inscribed quadrilateral opposite angles are in fact supplements for each other so[tex]m<M.A.F+m<M.E.F=180\°[/tex][tex]m<M.E.F=180\°-75\°=105\°[/tex]step 2Find the measure of arc M.A.Fwe know thatThe inscribed angle measures half that of the arc comprisingso[tex]m<M.E.F=\frac{1}{2}(arc\ M.A.F)[/tex] we have[tex]m<M.E.F=105\°[/tex]substitute[tex]105\°=\frac{1}{2}(arc\ M.A.F)[/tex][tex]arc\ M.A.F=210\°[/tex]step 3Find the measure of arc A.F[tex]arc\ M.A.F=arc\ A.M+arc\ A.F[/tex]we have[tex]arc\ M.A.F=210\°[/tex][tex]arc\ A.M=125\°[/tex]substitute[tex]210\°=125\°+arc\ A.F[/tex][tex]arc\ A.F=210\°-125\°=85\°[/tex]step 4Find the measure of angle A.M.Ewe know thatThe inscribed angle measures half that of the arc comprisingso[tex]m<A.M.E=\frac{1}{2}(arc\ A.F.E)[/tex]we have[tex]arc\ A.F.E=arc\ A.F+arc\ E.F=85\°+31\°=116\°[/tex]substitute[tex]m<A.M.E=\frac{1}{2}(116\°)=58\°[/tex]