Question

18.) givens: $overline{ae}$ bisects $overline{bd}$, $overline{ac}congoverline{ce}$. statements: $overline{ae}$ bisects $overline{bd}$ (given); (definition of bisector); $angle acbcongangle ecd$ (vertical angle theorem); $overline{ac}congoverline{ce}$ (given); $\triangle acbcong\triangle ecd$ (sas). reasons: given; definition of bisector; vertical angle theorem; given; sas.

Explanation:

Step1: State given fact

$\overline{AE}$ bisects $\overline{BD}$ (Given)

Step2: Use bisector definition

Let the intersection of $\overline{AE}$ and $\overline{BD}$ be point $F$. Then $BF = FD$ by the definition of a bisector.

Step3: Identify vertical - angles

$\angle ACB\cong\angle ECD$ because vertical angles are congruent.

Step4: State given side - congruence

$\overline{AC}\cong\overline{CE}$ (Given)

Step5: Prove triangle congruence

In $\triangle ACB$ and $\triangle ECD$, we have $\overline{AC}\cong\overline{CE}$, $\angle ACB\cong\angle ECD$, and since $\overline{AE}$ bisects $\overline{BD}$, we can show that the included sides and the included angle satisfy the Side - Angle - Side (SAS) congruence criterion. So, $\triangle ACB\cong\triangle ECD$ by SAS.

Answer:

The proof that $\triangle ACB\cong\triangle ECD$ is complete as shown above.