Question

use the given information to prove that $\triangle def \cong \triangle hgf$.
given: $\overline{ef} \cong \overline{gf}$
$\angle edf \cong \angle ghf$
prove: $\triangle def \cong \triangle hgf$
statement reason
1 $\overline{ef} \cong \overline{gf}$ reason?
2 $\angle edf \cong \angle \square$ given
3 $\angle dfe \cong \angle hfg$ reason?
4 $\triangle def \cong \triangle hgf$ reason?

Explanation:

Step1: Analyze Statement 1

The first statement is \(\overline{EF} \cong \overline{GF}\), and the reason is "Given" because this information is provided in the problem.

Step2: Analyze Statement 2

The second statement is \(\angle EDF \cong \angle \square\), and from the given information \(\angle EDF \cong \angle GHF\), so the angle should be \(\angle GHF\).

Step3: Analyze Statement 3

\(\angle DFE \cong \angle HFG\) because they are vertical angles, and vertical angles are congruent.

Step4: Analyze Statement 4

To prove \(\triangle DEF \cong \triangle HGF\), we use the AAS (Angle - Angle - Side) congruence criterion. We have \(\angle EDF \cong \angle GHF\) (given), \(\angle DFE \cong \angle HFG\) (vertical angles), and \(\overline{EF} \cong \overline{GF}\) (given), so by AAS, the triangles are congruent.

Answer:

1. Reason for \(\overline{EF} \cong \overline{GF}\): Given
2. \(\angle EDF \cong \angle \boldsymbol{GHF}\) (Given)
3. Reason for \(\angle DFE \cong \angle HFG\): Vertical angles are congruent
4. Reason for \(\triangle DEF \cong \triangle HGF\): AAS (Angle - Angle - Side) Congruence Criterion