Question

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

Explanation:

Step1: Recall vertical - angles property

$\angle AGF$ and $\angle EGB$ are vertical angles and by the Vertical Angles Theorem, $\angle AGF\cong\angle EGB$.

Step2: Recall corresponding - angles property

Since $\overline{AB}\parallel\overline{CD}$, by the Corresponding Angles Theorem, $\angle EGB\cong\angle EHD$.

Step3: Use transitive property

We know that if $\angle AGF\cong\angle EGB$ and $\angle EGB\cong\angle EHD$, then by the Transitive Property of Equality, $\angle AGF\cong\angle EHD$.

Answer:

$\angle AGF\cong\angle EHD$; Transitive Property of Equality