To prove that
[tex]\Delta ABC\cong\Delta BAD[/tex]
We have to prove that they share at least 2 angles.
1.
[tex]\angle D\cong\angle C[/tex]
This is a given fact.
2.
Notice that
[tex]\Delta ADE\cong\Delta\text{BEC}[/tex]
Since they already share two angles: DEA and CEB (They are vertically opposite)
This way, we can conclude that:
[tex]\angle DAE\cong\angle\text{CBE}[/tex]
In other words, the two angles on top of A and B are equal.
Therefore, we can conclude that
[tex]\angle DAB\cong\angle CBA[/tex]
And since ΔABC and ΔBAD share two of their angles, we can conclude that they also share their third and that:
[tex]\Delta ABC\cong\Delta BAD[/tex]
Q.E.D
