Question

answer part 1. * (10 points)
complete the proof.
given: ( overline { b d } ) bisects ( overline { a c } ) at point ( e ; angle b a e cong angle d c e ).
prove: ( \triangle a b e cong \triangle c d e )
we are given that ( overline { b d } ) bisects ( overline { a c } ) at point ( e ) and ( angle b a e cong angle d c e ).
because ( overline { b d } ) bisects ( overline { a c } ) at point ( e, overline { a e } cong overline { c e } ) by the
a. definition of a segment bisector b. segment addition postulate.
by the vertical angles theorem, ( angle a e b cong angle c e d ). therefore, ( \triangle a b e cong \triangle c d e )
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) postulate

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