Question

complete the proof that $overline{hi} parallel overline{fg}$.

	statement	reason
2	$overline{hi} \cong \overline{fg}$	given
3	$overline{fh} \cong \overline{fh}$	reflexive property of congruence
4	$\triangle fhi \cong \triangle hfg$	sss
5	$\angle fhi \cong \angle gfh$	cpctc
6	$overline{hi} \parallel \overline{fg}$

Explanation:

Step1: Identify the angles

We have $\angle FHI \cong \angle GFH$ from step 5 (CPCTC). These are alternate interior angles formed by the transversal $\overline{FH}$ intersecting $\overline{HI}$ and $\overline{FG}$.

Step2: Apply the Alternate Interior Angles Theorem

The Alternate Interior Angles Theorem states that if alternate interior angles are congruent, then the two lines cut by the transversal are parallel. Since $\angle FHI \cong \angle GFH$, by the Alternate Interior Angles Theorem, $\overline{HI} \parallel \overline{FG}$.

Answer:

Alternate Interior Angles Theorem