Question

Triangles F G H and F J H share common side F H. Sides F G and G H are congruent. Sides F J and J H are congruent.
Which explains whether ΔFGH is congruent to ΔFJH?

Answer 1

To determine whether triangles \( \Delta FGH \) and \( \Delta FJH \) are congruent, we can use the information that is provided.

1. **Common Side**: Both triangles share the side \( FH \).
2. **Congruent Sides**: We have that \( FG \cong GH \) and \( FJ \cong JH \).

Now, according to the information:

- In triangle \( \Delta FGH \):
- Side \( FH \) is common.
- We know \( FG \cong GH \).

- In triangle \( \Delta FJH \):
- Side \( FH \) is also common.
- We know \( FJ \cong JH \).

To prove that the triangles are congruent, we can use the Side-Side-Side (SSS) postulate, however, we currently lack the lengths for \( FG \) and \( FJ \) for a direct comparison.

Since we have the sides being congruent:

- If \( FG \cong FJ \) and \( GH \cong JH \) (this is inferred since \( GH \) and \( JH \) are congruent sides of the triangles), we can conclude that all corresponding sides of triangles \( FGH \) and \( FJH \) are congruent.

Thus, according to the Side-Side-Side (SSS) theorem, if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Hence, we can conclude that:

**Yes, triangles \( \Delta FGH \) and \( \Delta FJH \) are congruent by the SSS criterion, due to the congruent sides and common side \( FH \).**