A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal $\displaystyle 360$ degrees. And, the adjacent interior angles must be supplementary angles (sum of $\displaystyle 180$ degrees). 

Since, angles $\displaystyle E$ and $\displaystyle G$ are opposite interior angles, thus they must be equivalent. 

$\displaystyle \measuredangle E=72^\circ$, therefore $\displaystyle \measuredangle G=72^\circ$

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal $\displaystyle 360$ degrees. And, the adjacent interior angles must be supplementary angles (sum of $\displaystyle 180$ degrees).


Thus, the solution is:

$\displaystyle 180-72=108^\circ$

Since both angles $\displaystyle H$ and $\displaystyle F$ equal $\displaystyle 108.$ There sum must equal $\displaystyle 108+108=216^\circ$

A parallelogram must have equivalent opposite interior angles. Additionally, the sum of all four interior angles must equal $\displaystyle 360$ degrees.

Also, the adjacent interior angles must be supplementary angles (sum of $\displaystyle 180$ degrees).

Since, angles $\displaystyle F$ and $\displaystyle G$ are adjacent to each other they must be supplementary angles.

Thus, the sum of these two angles must equal $\displaystyle 180$ degrees. 

This question is very simple to answer if you remember that ALL paralellograms have two pairs of equal and opposite angles, and that the four angles in any quadrilateral MUST add up to 360 degrees. 

Because the angles given are different, we know that they are supplementary and the other two missing angles MUST be the same.

$\displaystyle 65^{\circ} + 115^{\circ} + 65^{\circ} + 115^{\circ} = 360^{\circ}$

Below is regular Pentagon $\displaystyle PENTA$ with center $\displaystyle C$, a segment drawn from $\displaystyle C$ to each vertex - that is, each of its radii drawn.

The measure of each angle of a regular pentagon can be calculated by setting $\displaystyle N$ equal to 5 in the formula

$\displaystyle \frac{(N-2)180^{\circ }}{N}$

and evaluating:

$\displaystyle \frac{(5-2)180^{\circ }}{5} = \frac{3 \cdot 180^{\circ }}{5} = \frac{540^{\circ }}{5} = 108^{\circ }$

Specifically, 

$\displaystyle m \angle PEN = 108 ^{\circ }$

By symmetry, each radius bisects one of these angles. Specifically, 

$\displaystyle m \angle CPE = \frac{1}{2} m \angle APE = \frac{1}{2} \cdot 108^{\circ } = 54 ^{\circ }$

By the Same-Side Interior Angles Theorem, consecutive angles of a parallelogram are supplementary - that is, their measures total $\displaystyle 180^{\circ }$. However,

$\displaystyle m \angle PEN + m \angle CPE = 108 ^{\circ } + 54 ^{\circ } = 162^{\circ }$,

violating these conditions. Therefore, Quadrilateral $\displaystyle CPEN$ is not a parallelogram.

$\displaystyle m \angle A + m \angle B = 80^{\circ } +100 ^{\circ } = 180^{\circ }$, making $\displaystyle \angle A$ and $\displaystyle \angle B$ supplementary. By the Converse of the Same Side Interior Angles Theorem, , it does follow that $\displaystyle \overline{AD} || \overline{BC}$. However, without knowing the measures of the other two angles, nothing further can be concluded about Quadrilateral $\displaystyle ABCD$. Below are a parallelogram and a trapezoid, both of which have these two angles of these measures.

By the Same-Side Interior Angles Theorem, consecutive angles of a parallelogram can be proved to be supplementary - that is, their angle measures total $\displaystyle 180^{\circ }$. Specifically, $\displaystyle \angle A$ and $\displaystyle \angle B$ are a pair of supplementary angles. Since they are also congruent, it follows that both are right angles. For the same reason, $\displaystyle \angle C$ and $\displaystyle \angle D$ are also right angles. The parallelogram, having four right angles, is a rectangle by definition.

The diagonals of a parallelogram are perpendicular - and, consequently, $\displaystyle \angle AXB$ is a right angle. - if and only if the parallelogram is a rhombus, a figure with four sides of equal length. Not all rectangles have four congruent sides.  Therefore, $\displaystyle \angle AXB$ need not be a right angle.

A rectangle is a parallelogram with four right angles.

Consecutive angles of a parallelogram are supplementary. If one angle of a parallelogram is given to be right, then its neighboring angles, being supplementary to a right angle, are right as well; also, opposite angles of a parallelogram are congruent, so the opposite angle is also right. All four angles must be right, making the parallelogram a rectangle by definition.