what is the reason for step 5 in this proof?
a. vertical angles theorem
b. opposite interior angles of a parallelogram are congruent.
c. all angles of a rectangle have the same measure.
d. alternate interior angles theorem
prove
$ \begin{array} { l } { \overline { g h } \cong \overline { e h } } \\ { \overline { d h } \cong \overline { f h } } \end{array} $
\( \begin{array} { | l | l | } \hline \text { statement } & \text { reason } \\ \hline \text { quadrilateral defg is a parallelogram } & \text { given } \\ \hline \overline { d e } \parallel \overline { g f } & \text { definition of a parallelogram } \\ \overline { d g } \parallel \overline { e f } & \\ \hline \text { draw } \overline { d f } \text { and } \overline { g e } \text { these line segments are } & \\ \text { transversals cutting two pairs of parallel lines } & \text { drawing line segments } \\ \text { de and } \overleftrightarrow { g f } \text { and } \overleftrightarrow { d g } \text { and } \overleftrightarrow { e f } & \\ \hline \text { place point } h \text { where } \overline { d f } \text { and } \overline { g e } \text { intersect } & \text { defining a point } \\ \hline \angle h g d \cong \angle h e f & \\ \angle h d g \cong \angle h f e & \\ \hline d g = e f & \text { opposite sides of a parallelogram } \\ & \text { are congruent } \\ \hline & \text { asa criterion for congruence } \\ \hline g h = e h & \text { corresponding sides of congruent } \\ d h = f h & \text { triangles are congruent } \\ \hline \end{array} \)

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# Brief Explanations:
- The Vertical Angles Theorem states that when two lines intersect, the opposite (vertical) angles are congruent.
- In a parallelogram, the Alternate Interior Angles Theorem is relevant when we have parallel sides and a transversal. Here, since $DE\parallel GF$ (definition of a parallelogram) and $DF$ is a transversal, $\angle HDG$ and $\angle HFE$ are alternate - interior angles.
- The Opposite interior angles of a parallelogram being congruent is not the case here as we are dealing with angles formed by a transversal and parallel sides.
- The figure is a parallelogram, not a rectangle, so the property of all angles of a rectangle having the same measure is not applicable.

# Answer:
D. Alternate Interior Angles Theorem

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