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)

By the ASA (Angle - Side - Angle) postulate, if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Answer:

B. ASA Postulate