Answer:
The measure of angle A.M.E is [tex]58\°[/tex]
Step-by-step explanation:
step 1
Find the measure of arc M.E
we know that
The inscribed angle is half that of the arc it comprises.
[tex]m<M.A.F=\frac{1}{2}(arc\ M.E+arc\ E.F)[/tex]
we have
[tex]m<M.A.F=75\°[/tex]
[tex]arc\ E.F=31\°[/tex]
substitute the values
[tex]75\°=\frac{1}{2}(arc\ M.E+31\°)[/tex]
[tex]150\°=(arc\ M.E+31\°)[/tex]
[tex]arc\ M.E=150\°-31\°=119\°[/tex]
step 2
Find the measure of arc A.F
we know that
[tex]arc\ M.E+arc\ E.F+arc\ A.F+arc\ A.M=360\°[/tex] -----> by complete circle
substitute the values
[tex]119\°+31\°+arc\ A.F+125\°=360\°[/tex]
[tex]119\°+31\°+arc\ A.F+125\°=360\°-275\°=85\°[/tex]
step 3
Find the measure of angle A.M.E
we know that
The inscribed angle is half that of the arc it comprises.
[tex]m<A.M.E=\frac{1}{2}(arc\ A.F+arc\ E.F)[/tex]
substitute the values
[tex]m<A.M.E=\frac{1}{2}(85\°+31\°)=58\°[/tex]
