Question

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

Explanation:

Step1: Identify Vertical Angles

$\angle DFE$ and $\angle GFH$ are vertical angles, so $\angle DFE \cong \angle GFH$ (Vertical Angles Theorem).

Step2: Apply SAS Congruence

We have:

• $\overline{EF} \cong \overline{HF}$ (Given)
• $\angle DFE \cong \angle GFH$ (Vertical Angles)
• $\overline{DF} \cong \overline{GF}$ (Given)

By the Side - Angle - Side (SAS) Congruence Postulate, $\triangle DEF \cong \triangle GHF$.

Answer:

$\triangle DEF \cong \triangle GHF$ by the SAS Congruence Postulate (using $\overline{EF} \cong \overline{HF}$, $\angle DFE \cong \angle GFH$, and $\overline{DF} \cong \overline{GF}$).