Let X be any point within a square ABCD. On AX, a square, AXYZ is described such that D is within it. Which one of the following is correct ?

Options

A. a. AX = DZ
B. b. `angle ADZ angle BAX`
C. c. `AD = DZ`
D. d. `BX = DZ`

Answer

To solve the problem, we need to analyze the configuration of the square ABCD and the square AXYZ described on line AX, where D is located within the square AXYZ. We will use geometric properties and relationships to derive the necessary conclusions.

### Step-by-Step Solution:

1. **Identify the Points and Squares**:
- Let ABCD be a square with vertices A, B, C, and D.
- Let X be any point inside the square ABCD.
- A square AXYZ is constructed on line segment AX such that D lies within the square AXYZ.

2. **Understanding the Configuration**:
- Since AXYZ is a square, all sides are equal, and all angles are right angles (90 degrees).
- The square AXYZ is oriented such that the line segment AX is one of its sides.

3. **Analyzing the Triangles**:
- Consider triangles ABX and ACZ.
- Triangle ABX is formed by points A, B, and X.
- Triangle ACZ is formed by points A, C, and Z.
- We will analyze the angles in these triangles.

4. **Angle Relationships**:
- Let angle BAX be θ.
- Then, angle XAD will be 90° - θ, since AX is a straight line and the angles on a straight line sum to 180°.
- For triangle AXYZ, angle ZAX is 90° because it is a square.

5. **Using Angle Sum Property**:
- In triangle AXYZ, we have:
- Angle ZAD + Angle XAD = 90°.
- Since angle XAD is 90° - θ, we can substitute:
- Angle ZAD + (90° - θ) = 90°.
- This simplifies to:
- Angle ZAD = θ.

6. **Establishing Congruence**:
- Now we have:
- Angle BAX = θ and Angle ZAD = θ.
- By the Angle-Angle (AA) criterion for triangle similarity, triangles ABX and ACZ are similar.

7. **Conclusion**:
- Since triangles ABX and ACZ are similar, the corresponding sides are proportional.
- Therefore, the length of BX is equal to the length of DJ (where D is the vertex of square AXYZ opposite to A).
- Hence, we conclude that BX = DJ.

### Final Answer:
The correct conclusion is that **BX = DJ**.