Question

statements reasons
\\( \angle b \cong \angle d \\)
\\( \overline { a c } \cong \overline { e c } \\) given
\\( \angle a c b \cong \angle e c d \\)
\\( \triangle a b c \cong \triangle e d c \\)
\\( \overline { a b } \cong \overline { e d } \\)

Explanation:

Step1: Reason for ∠ACB ≅ ∠ECD

Vertical angles are formed when two lines intersect. By the Vertical Angles Theorem, ∠ACB and ∠ECD (which are vertical angles) are congruent.

Step2: Reason for △ABC ≅ △EDC

We have ∠B ≅ ∠D (given), ∠ACB ≅ ∠ECD (from Step 1), and \(\overline{AC}\cong\overline{EC}\) (given). The AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent. So, △ABC ≅ △EDC by AAS.

Step3: Reason for \(\overline{AB}\cong\overline{ED}\)

Since △ABC ≅ △EDC, by the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, the corresponding sides \(\overline{AB}\) and \(\overline{ED}\) are congruent.

Answer:

1. Vertical Angles Theorem
2. AAS (Angle - Angle - Side)
3. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)