Question

fill in the missing statement and reason in the proof of the alternate interior angles theorem.

image of lines with points e, g, h, f collinear, and lines ab and cd parallel

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.

options:

• 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. First, recall the theorems related to parallel lines. When two parallel lines are cut by a transversal, corresponding angles are congruent (Corresponding Angles Theorem). Here, \(\overline{AB}\parallel\overline{CD}\) and \(EF\) is the transversal, so \(\angle EGB\) and \(\angle EHD\) are corresponding angles. So the first blank (reason) should be the Corresponding Angles Theorem, and the statement for that reason is \(\angle EGB\) and \(\angle EHD\) are congruent.
2. Then, we know \(\angle AGF\cong\angle EGB\) (vertical angles) and \(\angle EGB\cong\angle EHD\) (corresponding angles). By the Transitive Property of Equality, if \(a = b\) and \(b = c\), then \(a = c\). So substituting, \(\angle AGF\cong\angle EHD\). Looking at the options, the first option has the correct reason (Corresponding Angles Theorem) and then the correct conclusion using transitive property: \(\angle AGF\) and \(\angle EHD\) are congruent. The other options either use the wrong theorem (Alternate Exterior Angles Theorem is not applicable here as \(\angle EGB\) and \(\angle EHD\) are corresponding, not alternate exterior) or have incorrect statements.

Answer:

A. Corresponding Angles Theorem, \(\angle AGF\) and \(\angle EHD\) are congruent