Square Properties: Analyzing Congruent Triangles Formed by Diagonals

ABCD is a square.

Are the four triangles congruent?

To determine whether the four triangles are congruent, we will analyze each triangle in the square ABCD:

1. Identify the triangles: The diagonals of the square intersect at the center, forming four triangles: $\triangle AOD$, $\triangle BOC$, $\triangle COD$, and $\triangle AOB$.

2. Check diagonal properties: In a square, diagonals are equal in length and bisect each other at right angles. Thus, both diagonals AC and BD are equal and intersect at 90 degrees.

3. Analyze triangles using properties of the square:
- All sides of the square are equal, so the segments of the diagonals (e.g., AO, BO, CO, DO) are equal.
- The diagonals bisect each other, resulting in four equal segments: $AO = BO = CO = DO$.
- The angles at the center ($\angle AOB$, $\angle BOC$, $\angle COD$, $\angle DOA$) are each $90^\circ$ because diagonals of a square are perpendicular.

4. Use SAS congruence criterion: Each triangle shares the following properties:
- Two sides ($AO$ and $BO$ or equivalents) are equal for each triangle.
- The included angle between those sides is $90^\circ$.

Thus, by the SAS (Side-Angle-Side) criterion, the triangles $\triangle AOD$, $\triangle BOC$, $\triangle COD$, and $\triangle AOB$ are congruent.

Therefore, the solution to the problem is Yes.