Question

question
given: $overline{bd}$ and $overline{ac}$ bisect each other.
prove: $overline{ab}congoverline{cd}$.
step statement reason
1 $overline{bd}$ and $overline{ac}$ bisect each other given
2 $overline{ae}congoverline{ec}$ select a reason...
answer
you must answer all questions above in order to submit.

Explanation:

Step1: Definition of bisect

If a line - segment bisects another line - segment, it divides it into two congruent parts. Since $\overline{BD}$ and $\overline{AC}$ bisect each other, by the definition of bisecting line - segments, $\overline{AE}\cong\overline{EC}$.

Step2: Vertical angles are congruent

$\angle AEB\cong\angle DEC$ because they are vertical angles.

Step3: Definition of bisect

Also, since $\overline{BD}$ and $\overline{AC}$ bisect each other, $\overline{BE}\cong\overline{ED}$.

Step4: SAS (Side - Angle - Side) congruence criterion

In $\triangle AEB$ and $\triangle DEC$, we have $\overline{AE}\cong\overline{EC}$, $\angle AEB\cong\angle DEC$, and $\overline{BE}\cong\overline{ED}$. So, $\triangle AEB\cong\triangle DEC$ by the SAS congruence criterion.

Step5: Corresponding parts of congruent triangles are congruent (CPCTC)

Since $\triangle AEB\cong\triangle DEC$, then $\overline{AB}\cong\overline{CD}$ (corresponding parts of congruent triangles are congruent).

Answer:

$\overline{AB}\cong\overline{CD}$ is proven by using the definition of bisecting line - segments, vertical - angle congruence, the SAS congruence criterion, and CPCTC.