Question

If the measure of arc DEF is 248°, what is the measure of ∡DEF?

Answer 1

The figure attached has <DEF which is an inscribed angle. An inscribed angle is an angle with its vertex on the circle which is formed by two intersecting chords. According to formulas, the measurement of arc DF is twice the measurement of ∡DEF. Arc DF is equal to 360-248 or 112. Hence ∡DEF is equal to 56 degrees.

Answer 2

the complete question in the attached figure
we know that
 angle < DEF is an inscribed angle. 
An inscribed angle is an angle with its vertex on the circle which is formed by two intersecting chords. 
The inscribed angle measures half of the arc it comprises
the arc it comprises is the arc DCF
 the arc DCF is equal to

$(360-248)=112 degrees$

Hence
∡DEF is equal to  arc DCF/2
$(112/2)=56 degrees$

the answer is
∡DEF=$56$°