Question

given: \\(\overline{ef} \cong \overline{hf}\\) \\(\overline{df} \cong \overline{gf}\\) prove: \\(\triangle def \cong \triangle ghf\\) step statement reason 1 \\(\overline{ef} \cong \overline{hf}\\) given 2 \\(\overline{df} \cong \overline{gf}\\) given 3 \\(\triangle def \cong \triangle ghf\\) vertical angles property

Explanation:

Step1: Identify Angles

∠DFE and ∠GFH are vertical angles, so ∠DFE ≅ ∠GFH (Vertical Angles Theorem).

Step2: Apply SAS

We have EF ≅ HF (Given), DF ≅ GF (Given), and ∠DFE ≅ ∠GFH (Vertical Angles). So by SAS (Side - Angle - Side) Congruence Criterion, △DEF ≅ △GHF. The incorrect reason was using "Vertical Angles Property" for the triangle congruence; the correct reason is SAS.

Answer:

The correct reason for step 3 is "Side - Angle - Side (SAS) Congruence Criterion" (because we have two pairs of congruent sides and the included congruent angle from vertical angles, so SAS applies to prove the triangles congruent).