Final answer:
By using properties of isosceles triangles and the Angle-Side-Angle (ASA) postulate, we can prove that in isosceles triangle △ABE, if ∠ABC ≅ ∠AED, then segments BC and ED must be congruent.
Step-by-step explanation:
To prove that the segments BC and ED in an isosceles triangle △ABE are congruent when it's given that ∠ABC ≅ ∠AED, we can use the properties of isosceles triangles and the concept of congruent triangles.
Firstly, since △ABE is isosceles with base BE, we know that the legs AB and AE are congruent. We are given that ∠ABC ≅ ∠AED, which means that triangle ABC and triangle AED share a common side AB, and have two angles congruent. By the Angle-Side-Angle (ASA) postulate, these two triangles are congruent to each other.
Since the triangles are congruent, their corresponding parts are congruent. Thus, BC ≅ ED because they are corresponding sides of congruent triangles △ABC and △AED.