Use the definition of an angle bisector and find a common side for both triangles.
Statements
Reasons
1.
△ ABC is isosceles, EB bisects ∠ABC
1.
Given
2.
AB ≅ BC
2.
Definition of isosceles triangle
3.
∠ABE ≅ ∠CBE
3.
Definition of angle bisector
4.
BE ≅ BE
4.
Reflexive Property of Congruent Segments
5.
△ABE ≅ △CBE
5.
SAS Congruence Postulate
Practice makes perfect
We are given an isosceles triangle △ ABC where AB ≅ BC and also BE bisects ∠ABC, which means that ∠ABE ≅ ∠CBE.
Notice that BE is a common side for △ ABE and △ CBE, and by the Reflexive Property of Congruent Segments we get BE ≅ BE.
cc
AB ≅ BC & Side
∠ABE ≅ ∠CBE & Included Angle
BE ≅ BE & Side
By the Side-Angle-Side (SAS) Congruence Postulate we conclude that △ ABE ≅ △ CBE.
