Question

A square with vertices upper J, upper K, upper L, and upper M (in that order) that looks like a diamond. There are two dashed diagonals that intersect at point upper O.
If this quadrilateral JKLM
a square, identify all triangles that are congruent and explain why they are congruent.

Answer 1

To analyze the square JKLM with vertices J, K, L, and M (in that order), where the diagonals intersect at point O, we can identify and prove the congruence of several triangles formed within the square.

### Triangles Formed and Their Congruence

1. **Triangles JOK and LOM**
- **Identifying the triangles:**
- Triangle JOK consists of vertices J, O, and K.
- Triangle LOM consists of vertices L, O, and M.
- **Congruence Criteria:**
- Both triangles share the same side length:
- JK = LM (sides of the square are equal).
- JO = OL (both are half the length of the diagonal, which is common since diagonals of a square bisect each other).
- KO = OM (also both are half the length of the diagonal).
- Therefore, by the Side-Side-Side (SSS) congruence criterion, triangles JOK and LOM are congruent.

2. **Triangles JOM and KOL**
- **Identifying the triangles:**
- Triangle JOM consists of vertices J, O, and M.
- Triangle KOL consists of vertices K, O, and L.
- **Congruence Criteria:**
- Similar to the previous case:
- JM = KL (sides of the square are equal).
- JO = OK (each half the length of diagonal, common segment).
- OM = OL (again both are half the length of the diagonal).
- By the SSS congruence criterion, triangles JOM and KOL are congruent.

### Summary
Thus, from the analysis above, we have identified:
- Triangles **JOK** and **LOM** are congruent.
- Triangles **JOM** and **KOL** are congruent.

### Conclusion
In a square, the diagonals intersect at the midpoint and create angles and sides that are symmetric about the center, ensuring all congruences as detailed. Therefore, we can conclude that these triangles are congruent due to the properties of the square and the SSS congruence postulate.