Question

the proof that \\( \triangle a c b \cong \triangle e c d \\) is shown.
given: \\( \overline{a e} \\) and \\( \overline{d b} \\) bisect each other at \\( c \\).
prove: \\( \triangle a c b \cong \triangle e c d \\)
what is the missing statement in the proof?
\\( \angle b a c \cong \angle d e c \\)
\\( \angle a c d \cong \angle e c b \\)
\\( \angle a c b \cong \angle e c d \\)
\\( \angle b c a \cong \angle d c a \\)

Explanation:

Step1: Analyze vertical angles

When two lines intersect, vertical angles are formed. Here, lines \(AE\) and \(DB\) intersect at \(C\).
\(\angle ACB\) and \(\angle ECD\) are vertical angles.

Step2: Recall vertical - angle property

By the vertical - angle theorem, vertical angles are congruent. So, \(\angle ACB\cong\angle ECD\).

Answer:

\(\angle ACB\cong\angle ECD\) (the third option)