Question

1. fi = gj and ∠i = ∠g. complete the proof that △fgj ≅ △fif.

1 overline{fi} ≅ overline{gj}
2 ∠i ≅ ∠g
3 ∠f ≅ ∠f
4 △fgj ≅ △fif

Explanation:

Step1: Identify vertical angles

Since \(\angle FHG\) and \(\angle JHI\) are vertical angles, \(\angle FHG=\angle JHI\) (Vertical Angles Theorem).

Step2: Use AAS (Angle - Angle - Side) criterion

We are given \(HI = GJ\) (Statement 1) and \(\angle I=\angle G\) (Statement 2). Also, from Step 1 \(\angle FHG=\angle JHI\).
In \(\triangle GJF\) and \(\triangle IHE\), we have two angles (\(\angle G=\angle I\) and \(\angle FHG=\angle JHI\)) and a non - included side (\(HI = GJ\)).
By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle GJF\cong\triangle IHE\).

Step3: Corresponding parts of congruent triangles

Since \(\triangle GJF\cong\triangle IHE\), \(\angle E=\angle F\) (Corresponding Parts of Congruent Triangles are Congruent - CPCTC).

Step4: Prove \(\triangle EGJ\cong\triangle EIF\)

We know \(HI = GJ\), \(\angle E=\angle F\), and \(\angle I=\angle G\).
In \(\triangle EGJ\) and \(\triangle EIF\), using AAS ( \(\angle E=\angle F\), \(\angle G=\angle I\), \(GJ = HI\)), \(\triangle EGJ\cong\triangle EIF\) (AAS).

Answer:

1. Given; 2. Given; 3. CPCTC (Corresponding Parts of Congruent Triangles are Congruent); 4. AAS (Angle - Angle - Side)