Question

given: $overline{ab}paralleloverline{cd}$ and $overline{bd}$ bisects $overline{ac}$. prove: $\triangle abecong\triangle cde$. notes: quadrilateral properties are not permitted in this proof. step 1 statement $overline{ab}paralleloverline{cd}$, $overline{bd}$ bisects $overline{ac}$ reason given

Explanation:

Step1: Identify vertical angles

$\angle AEB=\angle CED$ (Vertical - angles are equal)

Step2: Use parallel - line property for alternate interior angles

Since $\overline{AB}\parallel\overline{CD}$, $\angle BAE=\angle DCE$ (Alternate interior angles are equal for parallel lines $\overline{AB}$ and $\overline{CD}$ with transversal $\overline{AC}$)

Step3: Use bisection property

Since $\overline{BD}$ bisects $\overline{AC}$, $AE = CE$ (Definition of a bisector)

Step4: Apply ASA congruence criterion

In $\triangle ABE$ and $\triangle CDE$, we have $\angle BAE=\angle DCE$, $AE = CE$, and $\angle AEB=\angle CED$. So, $\triangle ABE\cong\triangle CDE$ by the Angle - Side - Angle (ASA) congruence criterion.

Answer:

$\triangle ABE\cong\triangle CDE$ (by ASA)