**Proof that Triangle ABC is Congruent to Triangle CBE**

Given: Angle A = Angle B and AB = BC

To prove: Triangle ABC is congruent to Triangle CBE

Proof:

1. Draw a diagram of Triangle ABC and Triangle CBE.
2. Label the given information: Angle A = Angle B and AB = BC.
3. Draw a line through point B that is parallel to AC, and label the point where it intersects CE as D.
4. Since angle A = Angle B, and angle A + angle B + angle C = 180 degrees, we know that angle C = 90 degrees.
5. By drawing the line through point B parallel to AC, we know that angle DBC = angle B and angle BDC = angle C.
6. Since angle C = 90 degrees, angle BDC = 90 degrees - angle B.
7. Since AB = BC, we know that angle ABC = angle BCA.
8. By drawing the line through point B parallel to AC, we know that angle BCE = angle CBE.
9. Since angle BDC = 90 degrees - angle B, and angle ABC = angle BCA, we can say that angle BDC = angle ABC - angle BCE.
10. Substituting the values we know, we get angle 90 degrees - angle B = angle ABC - angle CBE.
11. Simplifying, we get angle CBE = angle B.
12. Therefore, Triangle ABC is congruent to Triangle CBE by Angle-Side-Angle congruence.

Conclusion:

Thus, we have proved that Triangle ABC is congruent to Triangle CBE by using the given information and the Angle-Side-Angle congruence rule.