Question

step statement reason
1 △bdf ≅ △cef
ad ≅ ae given
2 bd ≅ ce corresponding parts of congruent triangles are congruent
(cpcic)
3 ab ≅ ac substitution
4 ∠b ≅ ∠c substitution
5 ∠a ≅ ∠a reflexive property
6 △abe ≅ △acd sas

Explanation:

Step1: Use CPCTC

Since \(\triangle BDF\cong\triangle CEF\), by Corresponding Parts of Congruent Triangles are Congruent (CPCTC), we have \(\overline{BD}\cong\overline{CE}\).

Step2: Use substitution for \(\overline{AB}\cong\overline{AC}\)

We know that \(\overline{AB}=\overline{AD}+\overline{DB}\) and \(\overline{AC}=\overline{AE}+\overline{EC}\). Given \(\overline{AD}\cong\overline{AE}\) and \(\overline{BD}\cong\overline{CE}\) (from Step 1), by substitution \(\overline{AB}\cong\overline{AC}\).

Step3: Use substitution for \(\angle B\cong\angle C\)

In an isosceles triangle (since \(\overline{AB}\cong\overline{AC}\)), the base - angles are equal. We can also think of it in terms of triangle congruence and substitution of equal side - length relationships in angle - side relationships.

Step4: Use reflexive property

For any angle \(\angle A\), by the reflexive property of congruence, \(\angle A\cong\angle A\).

Step5: Use SAS (Side - Angle - Side)

We have \(\overline{AB}\cong\overline{AC}\) (from Step 2), \(\angle A\cong\angle A\) (from Step 4), and if we assume the side - angle - side (SAS) criterion for triangle congruence. If we consider \(\triangle ABE\) and \(\triangle ACD\), \(\overline{AB}\cong\overline{AC}\), \(\angle A\) is common, and if we can show the other pair of sides (using the given and previous congruences) \(\triangle ABE\cong\triangle ACD\) by SAS.

Answer:

The steps of the proof are justified as above.