Question

$\overline{hi} \parallel \overline{gj}$ and $\overline{gh} \parallel \overline{ij}$. complete the proof that $\triangle ghi \cong \triangle ijg$.

	statement	reason
2	$\overline{gh} \parallel \overline{ij}$
3	$\angle gih \cong \angle igj$
4	$\angle hgi \cong \angle gij$
5	$\overline{gi} \cong \overline{gi}$
6	$\triangle ghi \cong \triangle ijg$

Explanation:

Step1: Reason for Statement 1

Given that \(\overline{HI} \parallel \overline{GJ}\), so the reason is "Given".

Step2: Reason for Statement 2

Given that \(\overline{GH} \parallel \overline{IJ}\), so the reason is "Given".

Step3: Reason for Statement 3

Since \(\overline{HI} \parallel \overline{GJ}\) and \(\overline{GI}\) is a transversal, \(\angle GIH\) and \(\angle IGJ\) are alternate interior angles. So the reason is "Alternate Interior Angles Theorem".

Step4: Reason for Statement 4

Since \(\overline{GH} \parallel \overline{IJ}\) and \(\overline{GI}\) is a transversal, \(\angle HGI\) and \(\angle GIJ\) are alternate interior angles. So the reason is "Alternate Interior Angles Theorem".

Step5: Reason for Statement 5

\(\overline{GI}\) is a common side to both \(\triangle GHI\) and \(\triangle IJG\), so the reason is "Reflexive Property of Congruence".

Step6: Reason for Statement 6

In \(\triangle GHI\) and \(\triangle IJG\), we have \(\angle GIH \cong \angle IGJ\), \(\angle HGI \cong \angle GIJ\) and \(\overline{GI} \cong \overline{GI}\). By the Angle - Angle - Side (AAS) Congruence Criterion, \(\triangle GHI \cong \triangle IJG\). So the reason is "AAS Congruence Criterion".

Answer:

1. Reason: Given
2. Reason: Given
3. Reason: Alternate Interior Angles Theorem
4. Reason: Alternate Interior Angles Theorem
5. Reason: Reflexive Property of Congruence
6. Reason: AAS Congruence Criterion