Question

what is the missing reason in the proof?

1. ∠egf ≅ ∠hgk 1. vert. ∠s are ≅
2. kh ∥ ef 2. given
3. ∠f ≅ ∠k 3. alt. int. ∠s are ≅
4. g is the midpoint of kf 4. given
5. fg ≅ kg 5. def. of midpt.
6. △feg ≅ △khg 6. ?
7. hg ≅ eg 7. cpctc

sas
asa
aas
hl

Explanation:

Step1: Recall triangle congruence criteria

• \(SAS\) (Side - Angle - Side): Two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.
• \(ASA\) (Angle - Side - Angle): Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.
• \(AAS\) (Angle - Angle - Side): Two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle.
• \(HL\) (Hypotenuse - Leg): For right - angled triangles, the hypotenuse and a leg of one right - angled triangle are congruent to the hypotenuse and a leg of another right - angled triangle.

Step2: Analyze the given information

• We know \(\angle EGF\cong\angle HGK\) (from statement 1), \(\angle F\cong\angle K\) (from statement 3), and \(FG\cong KG\) (from statement 5).
• The side \(FG\) is between \(\angle EGF\) and \(\angle F\) in \(\triangle FEG\), and the side \(KG\) is between \(\angle HGK\) and \(\angle K\) in \(\triangle KHG\).

Answer:

ASA