Question

use the given information to prove that ( \triangle def cong \triangle 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 ) vertical angles property
4 ( \triangle def cong \triangle hgf ) asa congruence property
try one last time

Explanation:

Step1: Identify given information

We are given that \(\overline{DF}\cong\overline{HF}\) and \(\angle DEF\cong\angle HGF\).

Step2: Use vertical angles property

\(\angle DFE\) and \(\angle HFG\) are vertical angles. By the Vertical Angles Property, \(\angle DFE\cong\angle HFG\).

Step3: Apply ASA congruence

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

Answer:

\(\triangle DEF\cong\triangle HGF\) by ASA Congruence Property.