In two triangles ABC and FDE, `angleB=angleD=90^(@)`, AC = FE and BC = DE. Then `angleF` = ____.

Options

A. `angleC`
B. `angleA`
C. `angleB`
D. can't be determined

Answer

To solve the problem, we need to determine the measure of angle F in triangle FDE given the properties of triangles ABC and FDE.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have two triangles: ABC and FDE.
- Angle B = Angle D = 90° (both triangles have a right angle).
- AC = FE (the hypotenuses of both triangles are equal).
- BC = DE (one side of each triangle is equal).

2. **Draw the Triangles**:
- Sketch triangle ABC with B as the right angle.
- Sketch triangle FDE with D as the right angle.

3. **Label the Sides**:
- In triangle ABC, label the sides as follows:
- AB is one leg,
- BC is the other leg,
- AC is the hypotenuse.
- In triangle FDE, label the sides similarly:
- FD is one leg,
- DE is the other leg,
- FE is the hypotenuse.

4. **Apply the RHS Congruence Criterion**:
- We know that both triangles have a right angle (angle B and angle D).
- We have AC = FE (hypotenuses are equal).
- We have BC = DE (one side is equal).
- Therefore, by the RHS (Right angle-Hypotenuse-Side) criterion, triangle ABC is congruent to triangle FDE.

5. **Use CPCTC**:
- Since the triangles are congruent, corresponding angles are equal.
- Therefore, angle A in triangle ABC corresponds to angle F in triangle FDE.
- We need to find angle F.

6. **Determine the Measure of Angle F**:
- In triangle ABC, since angle B is 90°, the sum of the angles in triangle ABC is 180°.
- Thus, angle A + angle B + angle C = 180°.
- Since angle B = 90°, we have:
- angle A + 90° + angle C = 180°.
- This implies:
- angle A + angle C = 90°.
- Since angle F corresponds to angle A, we conclude:
- angle F = angle A.

7. **Conclusion**:
- Therefore, angle F = angle A, and since angle A + angle C = 90°, we can say that angle F is also equal to angle C.
- Hence, angle F = 90° - angle C.

### Final Answer:
Angle F = 90° (since angle A is complementary to angle C).