Question

answer part 1. * (10 points)
complete the proof.
given: ( overline{bd} ) bisects ( overline{ac} ) at point ( e ; angle bae cong angle dce ).
prove: ( \triangle abe cong \triangle cde )
we are given that ( overline{bd} ) bisects ( overline{ac} ) at point ( e ) and ( angle bae cong angle dce ).
because ( overline{bd} ) bisects ( overline{ac} ) at point ( e, overline{ae} cong overline{ce} ) by the a. definition of a segment bisector b. segment addition postulate.
by the vertical angles theorem, ( angle aeb cong angle ced ). therefore, ( \triangle abe cong \triangle cde ) by the a. aas theorem b. asa postulate c. sas postulate.

Explanation:

Step1: Analyze the given information

We know that \(BD\) bisects \(AC\) at point \(E\), so \(AE = CE\) (by the definition of a segment bisector). Also, \(\angle BAE=\angle DCE\) (given), and \(\angle AEB=\angle CED\) (vertical angles are congruent).

Step2: Apply the ASA (Angle - Side - Angle) criterion

In \(\triangle ABE\) and \(\triangle CDE\), we have \(\angle BAE=\angle DCE\) (angle), \(AE = CE\) (side), \(\angle AEB=\angle CED\) (angle).

Answer:

B. ASA Postulate