Supply the reasons missing from the proof shown below.
Given colon modifying above upper A upper B with bar congruent to modifying above upper A upper C with bar, angle upper B upper A upper D congruent to angle upper C upper A upper D Prove colon modifying above upper A upper D with bar bisects modifying above upper B upper C with bar
Reasons missing from the proof:
1. Given: Colon Modifying above A B with bar congruent to Modifying above A C with bar
2. Given: Angle B A D congruent to Angle C A D
3. Draw line segment A D
4. Assume: Modifying above A D with bar does not bisect Modifying above B C with bar
5. Therefore, there exists a point E on Modifying above B C with bar such that either AE is congruent to ED or DE is congruent to EC
6. Case 1: Assume AE is congruent to ED
7. Then, by the Side-Angle-Side (SAS) congruence, triangle AED is congruent to triangle CED
8. And by the angle bisector theorem, AE/EC = AD/DC
9. But we know AE = ED from assumption, so AD/DC = 1
10. This implies that AD is congruent to DC, contradicting the given information that Modifying above A B with bar is congruent to Modifying above A C with bar
11. Case 2: Assume DE is congruent to EC
12. Then, by the Side-Angle-Side (SAS) congruence, triangle ADE is congruent to triangle AEC
13. And by the angle bisector theorem, AD/DC = AE/EC
14. But we know DE = EC from assumption, so AD/DC = 1
15. This implies that AD is congruent to DC, contradicting the given information that Modifying above A B with bar is congruent to Modifying above A C with bar
16. Since both cases lead to a contradiction, our assumption that Modifying above A D with bar does not bisect Modifying above B C with bar must be false
17. Therefore, Modifying above A D with bar bisects Modifying above B C with bar