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} ).
reason?
2 ( angle def cong angle square ).
given
3 ( angle dfe cong angle hfg ).
reason?
4 ( \triangle def cong \triangle hgf ).
reason?

Explanation:

Step1: Identify vertical angles

Vertical angles are equal. So, $\angle DFE=\angle HFG$ (vertical angles theorem).

Step2: Check for ASA congruence

We have $DF = HF$ (given), $\angle DEF=\angle HGF$ (given), and $\angle DFE=\angle HFG$ (from step 1). By the Angle - Side - Angle (ASA) congruence criterion, $\triangle DEF\cong\triangle HGF$.

Answer:

1. Reason: Given
2. Statement: $\angle HGF$; Reason: Vertical Angles Theorem
3. Reason: ASA (Angle - Side - Angle) Congruence Criterion
4. Reason: ASA (Angle - Side - Angle) Congruence Criterion