Question

use the given information to prove that $\triangle def \cong \triangle hgf$.
given: $\overline{ef} \cong \overline{gf}$
$\overline{df} \cong \overline{hf}$
prove: $\triangle def \cong \triangle hgf$

statement	reason
2 $\overline{df} \cong \square$	given
3 $\angle dfe \cong \angle hfg$	reason?
4 $\triangle def \cong \triangle hgf$	reason?

Explanation:

Step1: Fill in the missing side for statement 2

From the given information, we know that \(\overline{DF} \cong \overline{HF}\), so we fill in \(HF\) for the blank in statement 2.

Step2: Determine the reason for statement 1

The reason for \(\overline{EF} \cong \overline{GF}\) is that it is given in the problem.

Step3: Determine the reason for statement 3

\(\angle DFE\) and \(\angle HFG\) are vertical angles, and vertical angles are congruent. So the reason is "Vertical angles are congruent".

Step4: Determine the reason for statement 4

We have two sides and the included angle congruent: \(\overline{EF} \cong \overline{GF}\), \(\overline{DF} \cong \overline{HF}\), and \(\angle DFE \cong \angle HFG\). This satisfies the Side - Angle - Side (SAS) congruence criterion for triangles. So the reason is "SAS (Side - Angle - Side) Congruence Criterion".

Answer:

1. Reason: Given
2. \(\overline{DF} \cong \boldsymbol{\overline{HF}}\)
3. Reason: Vertical angles are congruent
4. Reason: SAS (Side - Angle - Side) Congruence Criterion