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: $\triangle def \cong \triangle hgf$

(there is a diagram of two triangles $\triangle def$ and $\triangle hgf$ with $f$ being a common intersection point of their sides, and points $d$, $e$, $f$, $g$, $h$ labeled as shown.)

Explanation:

Step1: Identify Vertical Angles

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

Step2: List Given Congruences

$\overline{EF} \cong \overline{GF}$ (Given), $\angle EDF \cong \angle GHF$ (Given)

Step3: Apply AAS Congruence

In $\triangle DEF$ and $\triangle HGF$:

• $\angle EDF \cong \angle GHF$ (Given)
• $\angle DFE \cong \angle HFG$ (Vertical angles)
• $\overline{EF} \cong \overline{GF}$ (Given)

By Angle - Angle - Side (AAS) Congruence Criterion, $\triangle DEF \cong \triangle HGF$

Answer:

To prove $\triangle DEF \cong \triangle HGF$, we use the AAS (Angle - Angle - Side) congruence criterion:

Step 1: Identify Vertical Angles

• Statement: $\angle DFE \cong \angle HFG$
• Reason: Vertical angles are congruent.

Step 2: Recall Given Information

• Statement: $\overline{EF} \cong \overline{GF}$
• Reason: Given.

• Statement: $\angle EDF \cong \angle GHF$
• Reason: Given.

Step 3: Apply AAS Congruence

• Statement: $\triangle DEF \cong \triangle HGF$
• Reason: Angle - Angle - Side (AAS) Congruence Criterion (since $\angle EDF \cong \angle GHF$, $\angle DFE \cong \angle HFG$, and $\overline{EF} \cong \overline{GF}$).

The key steps rely on recognizing vertical angles and applying the AAS congruence rule with the given congruent angles and side.