$\overline{\mathit{M}\mathit{N}}$is the median of trapezoid $ZOID$.

If trapezoid ZOID is isosceles and localid="1638356865991" $\mathrm{m}\angle \mathrm{D}=80$, then localid="1638356856943" $\mathrm{m}\angle \mathrm{O}=\underset{¯}{?}$.

The given diagram is:

The theorem 5-18 states that the base angles of an isosceles trapezoid are congruent.

From the given diagram it can be noticed that the angles $\angle D$and $\angle I$are the base angles of an isosceles trapezoid.

Therefore by using the theorem 5-18 it can be said that the angles $\angle D$and $\angle I$are congruent.

Therefore, $\angle D\cong \angle I$.

Therefore, $\angle D=\angle I=80°$.

As, the trapezoid $ZOID$has parallel sides as $ZO$and $DI$.

As, $ZO\parallel DI$and $OI$is a transversal, therefore the angles $\angle O$and $\angle I$are the adjacent angles on the same side of transversal.

As, the sum of the adjacent angles is $180°$, therefore the sum of the angles $\angle O$and $\angle I$is $180°$.

Therefore, it can be obtained that:

$\begin{array}{c}\angle O+\angle I=180°\\ \angle O+80°=180°\\ \angle O=180°-80°\\ \angle O=100°\end{array}$

Therefore, the measure of the angle O is $100°$.

If trapezoid $ZOID$is isosceles and $m\angle D=80$, then $m\angle O=100$.