Question

1. developing proof complete the proof that \\( \angle b a c \cong \angle d b e \\).

given \\( b \\) is the midpoint of \\( \overline{a d} \\),
\\( \angle c \cong \angle e, \overline{b c} \\| \overline{d e} \\)
prove \\( \angle b a c \cong \angle d b e \\)

statements	reasons
2. \\( \overline{a b} \cong \overline{b d} \\)	2. \\( ? \\)
3. \\( \angle c \cong \angle e \\)	3. given
4. \\( \overline{b c} \\	\overline{d e} \\)	4. given
5. \\( \angle e d b \cong \angle c b a \\)	5. \\( ? \\)
6. \\( ? \\)	6. aas congruence theorem
7. \\( \angle b a c \cong \angle d b e \\)	7. \\( ? \\)

Explanation:

Step1: Midpoint Definition

By the definition of a midpoint, if \(B\) is the midpoint of \(\overline{AD}\), then \(AB = BD\), so \(\overline{AB}\cong\overline{BD}\).

Step2: Alternate Interior Angles

Since \(\overline{BC}\parallel\overline{DE}\), by the Alternate Interior Angles Theorem, \(\angle EDB\cong\angle CBA\).

Step3: AAS Congruence

By the AAS (Angle - Angle - Side) Congruence Theorem, \(\triangle ABC\cong\triangle BDE\).

Step4: CPCTC

If \(\triangle ABC\cong\triangle BDE\), then by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(\angle BAC\cong\angle DBE\).

Answer:

1. Definition of midpoint; 5. Alternate Interior Angles Theorem; 6. \(\triangle ABC\cong\triangle BDE\); 7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)