Triangles DEG and EGF are congruent by the ASA criterion, as angle DEG equals angle EGF, side EG is common, and angle GED equals angle GFE.
In the given quadrilateral DEFG, the triangles DEG and EGF are congruent due to the Angle-Side-Angle (ASA) congruence criterion. The key elements establishing this congruence are as follows:
Angle DEG = Angle EGF (Angle): Since DE is parallel to FG, corresponding angles DEG and EGF are alternate interior angles, making them congruent.
Side EG (Side): Diagonal EG serves as a common side for both triangles.
Angle GED = Angle GFE (Angle): Vertical angles GED and GFE are congruent as they share the same vertex.
By the ASA congruence criterion, triangles DEG and EGF are proven to be congruent. This means that all corresponding sides and angles of the triangles are equal, and the triangles can be superimposed upon each other.
In summary, triangles DEG and EGF are congruent due to the parallelism of DE and FG, establishing congruent alternate interior angles and sharing a common side, EG. This complies with the ASA congruence criterion, ensuring the triangles are identical.