Question

fill in the missing statement and reason in the proof of the alternate interior angles theorem. it is given that $overline{ab}$ is parallel to $overline{cd}$ and points e, g, h, and f are collinear. $angle agf$ and $angle egb$ are vertical and congruent by the vertical angles theorem. $angle egb$ and $angle ehd$ are congruent according to the ______. finally, ______ by the transitive property of equality. corresponding angles theorem, $angle agf$ and $angle ehd$ are congruent alternate exterior angles theorem, $angle egb$ and $angle ehd$ are congruent corresponding angles theorem, $angle egb$ and $angle ehd$ are congruent alternate exterior angles theorem, $angle agf$ and $angle ehd$ are congruent

Explanation:

1. $\angle EGB$ and $\angle EHD$ are corresponding angles when $\overline{AB}\parallel\overline{CD}$. By the Corresponding Angles Theorem, corresponding angles formed by parallel lines are congruent.
2. Since $\angle AGF\cong\angle EGB$ (Vertical - Angles Theorem) and $\angle EGB\cong\angle EHD$ (Corresponding Angles Theorem), by the Transitive Property of Equality, $\angle AGF\cong\angle EHD$.

Answer:

Corresponding Angles Theorem, $\angle AGF$ and $\angle EHD$ are congruent