https://file.aiquickdraw.com/mathbot/file/1728521284060ne5ytqrt.jpg

The problem in the image can be solved through angle relationships and properties of bisectors in geometry.

### Given:
1. ∠F is complementary to ∠FGJ.
2. ∠H is complementary to ∠HGJ.
3. GJ bisects ∠FGH.

### Conclusion:
We need to prove that ∠F ≅ ∠H (i.e., ∠F is congruent to ∠H).

### Solution:

1. Since GJ bisects ∠FGH, we know that:
   - ∠FGJ ≅ ∠HGJ (by the definition of an angle bisector, the bisected angles are congruent).

2. From the complementary angle relationships:
   - ∠F + ∠FGJ = 90° (since ∠F is complementary to ∠FGJ).
   - ∠H + ∠HGJ = 90° (since ∠H is complementary to ∠HGJ).

3. Since ∠FGJ ≅ ∠HGJ (from step 1), we substitute this into the complementary relationships:
   - ∠F + ∠HGJ = 90°.
   - ∠H + ∠HGJ = 90°.

4. Therefore, we can conclude that:
   - ∠F = ∠H (by subtracting ∠HGJ from both equations).

Thus, the conclusion ∠F ≅ ∠H is correct.

### Do you want any further details or clarifications?

### Related Questions:
1. How does the concept of complementary angles help in proving congruence in other geometric problems?
2. What other properties of angle bisectors can be used in solving geometric proofs?
3. How does congruence of angles relate to the congruence of triangles?
4. Can you apply the concept of bisectors in problems involving right triangles?
5. How would the conclusion change if GJ did not bisect ∠FGH?

### Tip:
When working with angle bisectors, always remember that the bisected angles are congruent, and this property is often key in proving further angle relationships.

Given: ∠F is complementary to ∠FGJ, ∠H is complementary to ∠HGJ, and GJ bisects ∠FGH. Conclusion: ∠F ≅ ∠H.

This problem explores how to prove that ∠F ≅ ∠H using the properties of angle bisectors and complementary angles. It involves bisecting angles and using complementary relationships to derive angle congruence, suitable for students in Grades 9-10.

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