Question

can $\triangle abe$ be proved congruent to $\triangle cde$ by the side - angle - side triangle congruency theorem from the figure? in the figure, point $e$ bisects both $\overline{ac}$ and $\overline{bd}$. if yes, select the appropriate pairs of corresponding parts to show that the triangles are congruent using the side - angle - side triangle congruency theorem. if no, select \cannot be proved.\ $\square \angle aeb \cong \angle ced$ $\square ab \cong cd$ $\square \angle eab \cong \angle ecd$ $\square ae \cong ce$ $\square \angle abe \cong \angle dce$ $\square be \cong de$ $\square$ cannot be proved

Explanation:

Step1: Analyze given info

Point \( E \) bisects \( \overline{AC} \) and \( \overline{BD} \), so \( AE = CE \) (bisector definition) and \( BE = DE \) (bisector definition). Also, \( \angle AEB \) and \( \angle CED \) are vertical angles, so \( \angle AEB \cong \angle CED \).

Step2: Apply SAS theorem

SAS requires two sides and the included angle. We have \( AE \cong CE \), \( \angle AEB \cong \angle CED \), and \( BE \cong DE \). So the corresponding parts are \( \angle AEB \cong \angle CED \), \( AE \cong CE \), \( BE \cong DE \).

Answer:

\( \angle AEB \cong \angle CED \), \( AE \cong CE \), \( BE \cong DE \)