Given that ABCD is a quadrilateral with ∠ A ≅ ∠ C and BC bisects ∠ ABC, we are asked to prove that ∠ ADB≅ ∠ CDB.
Let's first decide whether the given proof is valid.
BD≅BD by the Reflexive Property.
Since BD bisects ∠ ABC, it also bisects
∠ ADC. So ∠ ADB≅ CDB.
It starts with proving that BD is congruent to itself by the Reflexive Property of Congruence. This is a good start if we want to prove ∠ ADB≅ ∠ CDB by proving △ ADB≅ △ CDB.
However, in the second step, because BD bisects ∠ ABC, it is concluded that it also bisects ∠ ADC. If ABCD was a parallelogram, the second step would be reasonable, but we do not know if it is. Therefore, the given proof is not valid. Let's consider the fact that BD bisects ∠ ABC.
By the definition of an angle bisector, we can conclude that ∠ ABD is congruent to ∠ CBD.
∠ ABD ≅ ∠ CBD
We also know by the Reflexive Property of Congruence that BD is congruent to itself.
BD ≅ BD
Combining the above steps, we have that two angles and one side of △ ABD is congruent to the corresponding two angles and one side of △ BCD.
Therefore, by the Angle Angle Side (AAS) Congruence Theorem, △ ABD is congruent to △ BCD.
△ ABD ≅ △ BCD
Finally, since the corresponding parts of congruent triangles are congruent, we can conclude that ∠ ADB ≅ ∠ CDB.
Let's summarize the above process in a flow proof.
