The proof that ΔACB ≅ ΔECD is shown.

Question

Given: AE and DB bisect each other at C.
Prove: ΔACB ≅ ΔECD

Triangles A B C and C D E share common point C.

A flow chart has 5 boxes with arrows facing downward connecting the boxes. Each of the boxes are labeled. Box 1 contains line segment A E and line segment B E bisect each other at C and is labeled given. Box 2 contains line segment A C is-congruent-to line segment E C and is labeled definition of bisector. Box 3 contains question mark and is labeled vertical angles theorem. Box 4 contains line segment D C is-congruent-to line segment B C and is labeled definition of bisector. Box 5 contains triangle A C B is-congruent-to triangle E C D and is labeled SAS.

What is the missing statement in the proof?

∠BAC ≅ ∠DEC
∠ACD ≅ ∠ECB
∠ACB ≅ ∠ECD
∠BCA ≅ ∠DCA

Answer 1

To prove that triangles ΔACB ≅ ΔECD using the SAS (Side-Angle-Side) congruence criterion, we need to identify an angle that corresponds to the sides AC and EC, which are congruent by the definition of the bisector, and the sides BC and DC, which are also congruent by the corresponding definition.

In this case, since segments AE and DB bisect each other at C, that means they create vertical angles at point C. The angles at point C that are formed by lines AC and EC are the vertical angles formed with angles BAC and DEC.

Therefore, the missing statement that correctly fits Box 3 based on the provided options is:

∠BAC ≅ ∠DEC

This statement asserts that the angles BAC and DEC are congruent, allowing us to conclude by the SAS congruence criterion that ΔACB ≅ ΔECD.