Since we know PQRS is a parallelogram, we want to use the theorems about parallelograms to determine RS.

Opposite sides of a parallelogram are congruent so \overline{RS} \cong \overline{PQ}

RS = PQ = 2.41

The two labeled angles, \angle DEF and \angle EDG, are consecutive angles. Since DEFG is a parallelogram, the consecutive angles are supplementary.

We want to write an equation relating the two labeled angles and then solve for x.

Once we solve for x, we then want to use the theorem that states that opposite angles of a parallelogram are congruent. Using this theorem, we know that \angle DEF \cong \angle DGF.

We want to substitute the value we solved for x and into 5x and evaluate m \angle DEF as this will be the same as m \angle DGF.

Since we know that \angle DEF \cong \angle DGF, we know that \angle DGF = 5x

Substituting 25 for x and evaluating, we get 5(25)=125.

m\angle DGF = 125 \degree

We have many theorems that we can use to justify that a quadrilateral is a parallelogram. We need to decide which we want to use.

In this case, we have been told information about two of the four angles. If we knew some information about one of the other angles, we could use the parallelogram consecutive angles converse to decide if it is indeed a parallelogram or not.

Since we are told that \overline{AD} \parallel \overline{BC} we can use the Consecutive interior angles theorem to determine that \angle D and \angle C are supplementary.

Since \angle A \cong \angle C, this means that \angle D and \angle A are also supplementary.

Since we know that \angle D is supplementary to both of its consecutive angles, we have that the quadrilateral ABCD is a parallelogram using the parallelogram consecutive angles converse.