Question

use the given information to prove that △def≅△hgf.
given: \\( \overline{df} \cong \overline{hf} \\), \\( \angle def \cong \angle hgf \\)
prove: \\( \triangle def \cong \triangle hgf \\)
1 \\( \overline{df} \cong \overline{hf} \\) given
2 \\( \angle def \cong \angle hgf \\) given
3 \\( \angle dfe \cong \angle hfg \\) asa congruence property
4 \\( \triangle def \cong \triangle hgf \\) asa congruence property

Explanation:

Step1: Vertical angles are equal

$\angle DFE\cong\angle HFG$ (Vertical angles are congruent)

Step2: Use ASA congruence

In $\triangle DEF$ and $\triangle HGF$, we have $\overline{DF}\cong\overline{HF}$ (given), $\angle DEF\cong\angle HGF$ (given), $\angle DFE\cong\angle HFG$ (vertical angles). By the ASA (Angle - Side - Angle) congruence property, $\triangle DEF\cong\triangle HGF$.

Answer:

1. $\overline{DF}\cong\overline{HF}$ - Given
2. $\angle DEF\cong\angle HGF$ - Given
3. $\angle DFE\cong\angle HFG$ - Vertical angles
4. $\triangle DEF\cong\triangle HGF$ - ASA Congruence Property