https://file.aiquickdraw.com/mathbot/file/1727216521330y8d2mkag.jpg

Let's complete the two-column proof in the image you uploaded.

### Given:
- ∠ABC and ∠HGF are complementary.
- \( m∠ABC = 71^\circ \).

### To Prove:
- \( m∠HGF = 19^\circ \).

### Two-Column Proof:

| **Statement**                                     | **Reason**                               |
|--------------------------------------------------|------------------------------------------|
| 1. ∠ABC and ∠HGF are complementary               | Given                                    |
| 2. \( m∠ABC + m∠HGF = 90^\circ \)                | Definition of complementary angles       |
| 3. \( m∠ABC = 71^\circ \)                        | Given                                    |
| 4. \( 71^\circ + m∠HGF = 90^\circ \)             | Substitution Property                    |
| 5. \( m∠HGF = 19^\circ \)                        | Subtraction Property (from 90°)          |

This completes the proof showing that \( m∠HGF = 19^\circ \).

Would you like more details on any of these steps or have any questions?

Here are five related questions to expand on the concept:
1. What are supplementary angles and how do they differ from complementary angles?
2. Can complementary angles ever be non-adjacent?
3. What is the sum of the interior angles in a triangle, and how does this relate to complementary angles?
4. How can you use complementary angles to solve for unknown angles in right triangles?
5. What is the relationship between angles in parallel lines and transversals?

**Tip:** Complementary angles always sum to 90°, which can help solve for unknown angle measures when one angle is given.

Complete the two-column proof for complementary angles: Given ∠ABC and ∠HGF are complementary and m∠ABC = 71°, prove that m∠HGF = 19°.

This geometry problem demonstrates how to complete a two-column proof to show that ∠ABC and ∠HGF are complementary. Given that m∠ABC = 71°, use key geometric properties to prove that m∠HGF = 19°. Suitable for students in Grades 8-10.

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