To determine whether triangles $\triangle WIN$ and $\triangle WND$ are congruent, let's analyze the given information in the diagram:

1. Shared Side (WN): Both triangles $\triangle WIN$ and $\triangle WND$ share side $WN$. This means that $WN$ is congruent to itself by the Reflexive Property.

2. Angle Markings (Base Angles): Both triangles have congruent base angles at $I$ and $D$, as indicated by the identical angle markings at these points. We can assume $\angle I \cong \angle D$.

3. Vertical Angles (at Vertex W): The two triangles also have a common vertex at $W$, where they form vertical angles across the line segment $ND$. Therefore, $\angle WIN \cong \angle WND$ by the Vertical Angles Theorem.

Conclusion

Since we have:

• $WN$ congruent to itself (shared side),
• $\angle I \cong \angle D$ (corresponding base angles),
• $\angle WIN \cong \angle WND$ (vertical angles),

we can use the Angle-Angle-Side (AAS) Congruence Theorem to conclude that $\triangle WIN \cong \triangle WND$.

Answer

So the congruence statement and theorem that proves it would be:

$\triangle WIN \cong \triangle WND \quad \text{by AAS}$

Would you like further details on the reasoning behind this or any additional explanation?

Here are some related questions you might find helpful:

1. What other congruence theorems could be applicable if we had additional information?
2. How does the Reflexive Property help in proving congruence between triangles?
3. What is the difference between AAS and ASA congruence theorems?
4. Can two triangles be congruent if they share no sides but have congruent angles?
5. How would the solution change if only one angle pair was congruent?

Tip: Always look for shared sides or angles in diagrams, as they often provide useful clues for proving congruence.