Question

the proof that δacb ≅ δecd is shown. given: $overline{ae}$ and $overline{db}$ bisect each other at c. prove: δacb ≅ δecd. what is the missing statement in the proof? options: ∠bac ≅ ∠dec, ∠acd ≅ ∠ecb, ∠acb ≅ ∠ecd, ∠bca ≅ ∠dca

Explanation:

To prove \(\triangle ACB \cong \triangle ECD\), we use the given that \(\overline{AE}\) and \(\overline{DB}\) bisect each other at \(C\), so \(AC = EC\) and \(BC = DC\) (by definition of bisecting). We need a pair of congruent angles between these sides. \(\angle ACB\) and \(\angle ECD\) are vertical angles, and vertical angles are congruent. Let's analyze the options:

• \(\angle BAC \cong \angle DEC\): These are not necessarily given or vertical angles, so not the missing statement.
• \(\angle ACD \cong \angle ECB\): These are not the angles between the sides we have ( \(AC, BC\) and \(EC, DC\) ).
• \(\angle ACB \cong \angle ECD\): These are vertical angles, so they are congruent. This is the angle needed for SAS (Side - Angle - Side) congruence (since \(AC = EC\), \(\angle ACB=\angle ECD\), and \(BC = DC\)).
• \(\angle BCA \cong \angle DCA\): These angles are not related to the sides of the two triangles we are proving congruent.

Answer:

\(\boldsymbol{\angle ACB \cong \angle ECD}\)