Answer:  The measure of ∠E is 75°.
Step-by-step explanation:  Given that DEFG is an isosceles trapezoid, where m∠D = 105° and m∠F = 75°.
We are to find the measure of ∠E.
ISOSCELES TRAPEZOID is a trapezoid in which the base angles are equal and so the left and right side lengths are also equal.
Therefore, in the isosceles trapezoid DEFG, we have
m∠D = m∠G,  m∠E = m∠F  and  DE = FG.
Since, m∠D = 105°, so m∠G =  105°.
The sum of all the four angles of a quadrilateral is 360° and a trapezoid is a quadrilateral with a set of opposite parallel sides, so in the trapezoid DEFG, we have

Thus, the measure of ∠E is 75°.