Question

ngle congruence
the proof that \\( \overline { h g } \cong \overline { e g } \\) is shown.
given: \\( g \\) is the midpoint of \\( \overline { k f } \\)
\\( \overline { k h } \parallel \overline { e f } \\)
prove: \\( \overline { h g } \cong \overline { e g } \\)

1. \\( g \\) is the midpoint of \\( k f ^ { prime } \\) 4. given
2. \\( \overline { f g } \cong \overline { k g } \\) 5. def. of midpt.
3. \\( \triangle f e g \cong \triangle k h g \\) 6. ?
4. \\( \overline { h g } \cong \overline { e g } \\) 7. cpctc

aas
asa
sas
hl

Explanation:

Step1: Identify vertical angles

$\angle EGF\cong\angle HGK$ (vertical angles are congruent)

Step2: Use parallel lines to find congruent angles

Since $KH\parallel EF$, $\angle K\cong\angle F$ (alternate - interior angles)

Step3: Check congruent triangles

We have $\overline{FG}\cong\overline{KG}$ (from the mid - point definition), $\angle EGF\cong\angle HGK$, $\angle K\cong\angle F$. So, by the AAS (Angle - Angle - Side) congruence criterion, $\triangle FEG\cong\triangle KHG$

Answer:

AAS