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The image shows a proof table where we need to complete the reasoning to show that \( \angle J \cong \angle K \). Let's complete the proof.

**Given:**
- \( \angle K \) and \( \angle I \) are complementary.
- \( \angle J \) and \( \angle I \) are complementary.

### Proof Steps:

| **Statement** | **Reason** |
|---------------|------------|
| 1. \( \angle K \) and \( \angle I \) are complementary | Given |
| 2. \( \angle J \) and \( \angle I \) are complementary | Given |
| 3. \( m\angle K + m\angle I = 90^\circ \) | Definition of complementary angles |
| 4. \( m\angle J + m\angle I = 90^\circ \) | Definition of complementary angles |
| 5. \( m\angle K + m\angle I = m\angle J + m\angle I \) | Transitive Property of Equality (both sums equal 90°) |
| 6. \( m\angle J = m\angle K \) | Properties of addition, subtraction (subtract \( m\angle I \) from both sides) |
| 7. \( \angle J \cong \angle K \) | Definition of congruent angles (equal measures) |

This completes the proof that \( \angle J \cong \angle K \).

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Would you like further details or explanations on any part of this proof? Here are a few related questions that could deepen understanding:

1. Why are angles considered complementary when their sum is 90 degrees?
2. Can the transitive property of equality be applied in cases other than equality of angles?
3. How is the definition of congruent angles connected to their measures?
4. Could a similar approach be used to prove angles are supplementary instead of complementary?
5. What happens if two angles are neither complementary nor supplementary?

**Tip:** When proving congruence or equality of angles, always ensure that the definitions of complementary, supplementary, and congruent angles are clearly understood.

Complete the proof that \( \angle J \cong \angle K \) given that \( \angle K \) and \( \angle I \) are complementary and \( \angle J \) and \( \angle I \) are complementary.

This step-by-step proof demonstrates how to show that two angles are congruent based on complementary relationships. Learn how the transitive property and definitions of complementary and congruent angles are applied in this geometry problem, suitable for students in Grades 7-10.

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