Given the triangles shown in the picture, you can identify that the included angles are corresponding and congruent and there is a pair of corresponding sides that are congruent too.
By definition, you can use five theorems to determine if two triangles are congruent:
1. SSS (Side-Side-Side): this states that if all the corresponding sides of the triangles have the same length, the triangles are congruent.
2. SAS (Side-Angle-Side): this states that if the included angle of the triangles are equal and the two included sides are equal, then the triangles are congruent.
3. ASA (Side-Angle-Side): if two sides and the included angle of both triangles are the same, then they are congruent.
4. AAS (Angle-Angle-Side): If the two included angles of both triangles are the same and one included side of both triangles are the same, then the triangles are congruent.
5. HL (Hypotenuse, Leg): this states that if the hypotenuse and one of the legs of both Right Triangles are the same, then the triangles are congruent.
In this case, you can identify that the triangles are not Right Triangles and you only know that they have a congruent side and a congruent angle. Therefore, you cannot determine their congruency using the theorems shown before.
Hence, the answer is: They cannot be proven congruent by SSS, SAS, ASA, AAS or HL.