Question

Must quad. EFGH be a parallelogram? Can it be a parallelogram? Explain.

Short answer

EFGH is a parallelogram.

Step-by-step solution

Step 1. Properties of a parallelogram

It is known that a parallelogram is a quadrilateral whose both pairs of opposite sides are parallel.

From the above figure, it can be observed that the angles $\angle XHG$and$\angle HEF$ are the corresponding angles that are formed when the lines $\overline{HG}$and$\overline{EF}$ are cut by the transversal $\overline{XE}$.

Step 2. Check the pair of parallel lines

It can be observed that,

$\angle XHG=\angle HEF=70°$

That is,

$\angle XHG\cong \angle HEF$

Thus, by postulate 11 which states that the relation between two lines is parallel only when a separate line travels through these two lines and the angles that are corresponding will be congruent, it can be said that the lines $\overline{HG}$and$\overline{EF}$ are parallel.

Step 3. Check whether EFGH is a parallelogram

The angles $\angle GFY$and $\angle HEF$are the corresponding angles that are formed when the lines $\overline{HE}$and $\overline{GF}$are cut by the transversal $\overline{EY}$.

It can be observed that,

$\angle GFY=\angle HEF=70°$

That is, $\angle GFY\cong \angle HEF$

Thus, by postulate 11 which states that the relation between two lines is parallel only when a separate line travels through these two lines and the angles that are corresponding will be congruent, it can be said that the lines $\overline{HE}$and $\overline{GF}$are parallel.

That is, for the quadrilateral EFGH whose both pairs of opposite sides are parallel.

So, it can be said that EFGH is a parallelogram.