Question

for items 11 - 13, refer to the diagram. if $overline{bd}$ bisects $overline{ac}$ and $overline{ac}$ bisects $overline{bd}$, is $\triangle abecong\triangle cde$? explain. a. yes; the triangles are congruent by side - side - angle. b. yes; the triangles are congruent by side - side - side. c. yes; the triangles are congruent by angle - angle - side. d. no; the triangles are not congruent.

Explanation:

Step1: Recall bisecting - property

If $\overline{BD}$ bisects $\overline{AC}$ and $\overline{AC}$ bisects $\overline{BD}$, then $AE = CE$ and $BE=DE$.

Step2: Identify vertical - angles

$\angle AEB$ and $\angle CED$ are vertical - angles, so $\angle AEB=\angle CED$.

Step3: Apply congruence - criterion

In $\triangle ABE$ and $\triangle CDE$, we have $AE = CE$, $\angle AEB=\angle CED$, and $BE = DE$. By the Side - Angle - Side (SAS) congruence criterion, $\triangle ABE\cong\triangle CDE$. But among the given options, the closest correct reasoning is based on the fact that we have two pairs of equal sides ($AE = CE$ and $BE = DE$) and the included angles (vertical angles) are equal. If we consider the order of the options and the concept of congruence, we know that the triangles are congruent.

Answer:

A. Yes; the triangles are congruent by Side - Angle - Side (even though it's not exactly as stated in the options, the triangles are congruent and the closest correct reasoning is in option A considering the equal sides and included angle formed by the bisection and vertical - angle properties).