Which of the following statement(s) is/are CORRECT about a triangle?

Options

A. If the triangles are congruent, their area will be equal.
B. If the areas of two similar triangles are equal, the triangles are congruent.
C. Two congruent triangles are always similar to each other
D. All options are correct

Answer

To determine which statements about triangles are correct, we need to analyze the properties of congruence and similarity in triangles. Let's break down the statements step by step.

### Step-by-Step Solution:

1. **Understanding Similar Triangles**:
- Two triangles are similar if their corresponding angles are equal. This means if triangle PQR has angles equal to triangle ABC, then PQR ~ ABC (they are similar).
- **Hint**: Remember that similarity is based on angle equality, not side lengths.

2. **Understanding Congruent Triangles**:
- Two triangles are congruent if they are exactly the same in shape and size. This can be established through several criteria:
- Side-Side-Side (SSS): All three sides of one triangle are equal to the corresponding sides of another triangle.
- Side-Angle-Side (SAS): Two sides and the angle between them in one triangle are equal to the corresponding parts of another triangle.
- Angle-Side-Angle (ASA): Two angles and the side between them in one triangle are equal to the corresponding parts of another triangle.
- Right angle-Hypotenuse-Side (RHS): In right triangles, the hypotenuse and one side are equal.
- **Hint**: Congruence requires exact matches in both size and shape.

3. **Area of Congruent Triangles**:
- If two triangles are congruent, their areas will also be equal. This is because congruent triangles cover the same amount of space.
- **Hint**: Congruent triangles have equal areas.

4. **Area of Similar Triangles**:
- If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. However, if the areas of two similar triangles are equal, they must be congruent.
- **Hint**: Equal areas in similar triangles imply that they are congruent.

5. **Conclusion on Statements**:
- **Statement 1**: If the triangles are congruent, then their areas are equal. (Correct)
- **Statement 2**: If the area of two similar triangles is equal, then the triangles are congruent. (Correct)
- **Statement 3**: Two congruent triangles are always similar. (Correct)
- **Statement 4**: All options are correct. (Correct)

### Final Answer:
All the statements provided about triangles are correct. Therefore, the correct option is **D: All options are correct**.