Chapter 5.1, Problem 6CE

To determine

To verify: if $EFGH$ is a parallelogram or not.

Answer to Problem 6CE

  $EFGH$ is a parallelogram.

Explanation of Solution

Given:

The diagram:

Consider the figure shown below:

It is known that a parallelogram is a quadrilateral whose both pair 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 which are formed when the lines $\overline{HG}$ and $\overline{EF}$ are cut by the transversal $\overline{XE}$ .

It can be observed that,

  $\angle XHG=\angle HEF=70°$

That is,

  $\angle XHG\cong \angle HEF$

Thus, by the postulate 11, it can be said that the lines $\overline{HG}$ and $\overline{EF}$ are parallel.

Similarly, the angles $\angle GFY$ and $\angle HEF$ are the corresponding angles which 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 the postulate 11, it can be said that the lines $\overline{HE}$ and $\overline{GF}$ are parallel.

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

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