Isosceles trapezoid angle calculation

In an isosceles trapezoid, base angles are equal. Since $\angle F = 59^{\circ}$, then $\angle E = 59^{\circ}$. The sum of angles in a quadrilateral is $360^{\circ}$. Thus, $\angle D + \angle G + \angle E + \angle F = 360^{\circ}$. Given $\angle D = 121^{\circ}$, we have $121^{\circ} + \angle G + 59^{\circ} + 59^{\circ} = 360^{\circ}$. Solving for $\angle G$: $\angle G = 360^{\circ} - 121^{\circ} - 59^{\circ} - 59^{\circ} = 121^{\circ}$. However, in an isosceles trapezoid, consecutive angles between parallel sides are supplementary. If DG is parallel to EF, then $\angle D + \angle E = 180^{\circ}$ and $\angle G + \angle F = 180^{\circ}$. Given $\angle D = 121^{\circ}$, then $\angle E = 180^{\circ} - 121^{\circ} = 59^{\circ}$. This matches the given $\angle F = 59^{\circ}$. Therefore, $\angle G = 180^{\circ} - \angle F = 180^{\circ} - 59^{\circ} = 121^{\circ}$. If EF is parallel to DG, then $\angle E + \angle D = 180^{\circ}$ and $\angle F + \angle G = 180^{\circ}$. Given $\angle D = 121^{\circ}$, then $\angle E = 180^{\circ} - 121^{\circ} = 59^{\circ}$. This matches the given $\angle F = 59^{\circ}$. Therefore, $\angle G = 180^{\circ} - \angle F = 180^{\circ} - 59^{\circ} = 121^{\circ}$. The diagram shows $\angle D$ and $\angle F$ as adjacent angles. In an isosceles trapezoid, adjacent angles between the parallel sides are supplementary. If DG is parallel to EF, then $\angle D + \angle E = 180^{\circ}$ and $\angle G + \angle F = 180^{\circ}$. Given $\angle D = 121^{\circ}$, then $\angle E = 180^{\circ} - 121^{\circ} = 59^{\circ}$. Since $\angle F = 59^{\circ}$, this is consistent. Therefore, $\angle G = 180^{\circ} - \angle F = 180^{\circ} - 59^{\circ} = 121^{\circ}$. However, the diagram indicates that $\angle D$ and $\angle F$ are not adjacent angles on the same base. In an isosceles trapezoid, the angles at each base are equal. If EF is the longer base, then $\angle E = \angle F = 59^{\circ}$. Then $\angle D = \angle G$. The sum of angles is $360^{\circ}$, so $2\angle D + 2\angle E = 360^{\circ}$. $2\angle D + 2(59^{\circ}) = 360^{\circ}$. $2\angle D = 360^{\circ} - 118^{\circ} = 242^{\circ}$. $\angle D = 121^{\circ}$. This matches the given $\angle D$. Therefore, $\angle G = \angle D = 121^{\circ}$. If DG is the longer base, then $\angle D = \angle G = 121^{\circ}$. Then $\angle E = \angle F$. The sum of angles is $360^{\circ}$, so $2\angle D + 2\angle E = 360^{\circ}$. $2(121^{\circ}) + 2\angle E = 360^{\circ}$. $242^{\circ} + 2\angle E = 360^{\circ}$. $2\angle E = 360^{\circ} - 242^{\circ} = 118^{\circ}$. $\angle E = 59^{\circ}$. This matches the given $\angle F = 59^{\circ}$. Therefore, $\angle G = 121^{\circ}$. The diagram shows $\angle D$ and $\angle F$ as consecutive angles. In an isosceles trapezoid, angles on the same base are equal. If EF is a base, then $\angle E = \angle F = 59^{\circ}$. Then the other base angles are $\angle D$ and $\angle G$. Since it is an isosceles trapezoid, $\angle D = \angle G$. The sum of angles in a quadrilateral is $360^{\circ}$. So, $\angle D + \angle G + \angle E + \angle F = 360^{\circ}$. $\angle D + \angle D + 59^{\circ} + 59^{\circ} = 360^{\circ}$. $2\angle D + 118^{\circ} = 360^{\circ}$. $2\angle D = 360^{\circ} - 118^{\circ} = 242^{\circ}$. $\angle D = 121^{\circ}$. This matches the given $\angle D$. Therefore, $\angle G = \angle D = 121^{\circ}$. However, the diagram shows $\angle D = 121^{\circ}$ and $\angle F = 59^{\circ}$. In an isosceles trapezoid, consecutive angles between parallel sides are supplementary. If DG is parallel to EF, then $\angle D + \angle E = 180^{\circ}$ and $\angle G + \angle F = 180^{\circ}$. Given $\angle D = 121^{\circ}$, then $\angle E = 180^{\circ} - 121^{\circ} = 59^{\circ}$. This matches $\angle F = 59^{\circ}$. Therefore, $\angle G = 180^{\circ} - \angle F = 180^{\circ} - 59^{\circ} = 121^{\circ}$. The diagram shows $\angle D$ and $\angle F$ as angles on opposite bases. In an isosceles trapezoid, angles on the same base are equal. If EF is a base, then $\angle E = \angle F = 59^{\circ}$. Then $\angle D = \angle G$. The sum of angles is $360^{\circ}$. $\angle D + \angle G + \angle E + \angle F = 360^{\circ}$. $\angle D + \angle D + 59^{\circ} + 59^{\circ} = 360^{\circ}$. $2\angle D + 118^{\circ} = 360^{\circ}$. $2\angle D = 242^{\circ}$. $\angle D = 121^{\circ}$. This matches the given $\angle D$. Therefore, $\angle G = 121^{\circ}$. Let's re-examine the diagram. It appears that DG is parallel to EF. In an isosceles trapezoid, angles on the same base are equal. If EF is a base, then $\angle E = \angle F = 59^{\circ}$. Then $\angle D$ and $\angle G$ are angles on the other base. Since it's an isosceles trapezoid, $\angle D = \angle G$. The sum of angles in a quadrilateral is $360^{\circ}$. So, $\angle D + \angle G + \angle E + \angle F = 360^{\circ}$. $\angle D + \angle D + 59^{\circ} + 59^{\circ} = 360^{\circ}$. $2\angle D + 118^{\circ} = 360^{\circ}$. $2\angle D = 360^{\circ} - 118^{\circ} = 242^{\circ}$. $\angle D = 121^{\c...