Geometry two-column proof example with labeled angles and statements.

Let's complete the two-column proof step by step, based on the given information and diagram.

We are given:

- $ m\angle 1 = m\angle 3 $
- We need to prove: $ m\angle EBA = m\angle DBC $

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Step-by-step Explanation:

From the diagram:
- Point B is the vertex.
- Angles are labeled:
- $ \angle 1 = \angle CBE $
- $ \angle 2 = \angle EBD $
- $ \angle 3 = \angle DBA $

So:
- $ \angle EBA = \angle EBD + \angle DBA = \angle 2 + \angle 3 $
- $ \angle DBC = \angle DBE + \angle EBC = \angle 2 + \angle 1 $

We are told $ m\angle 1 = m\angle 3 $. So if we can show both $ \angle EBA $ and $ \angle DBC $ equal $ \angle 2 + \angle 3 $ (or $ \angle 2 + \angle 1 $), then they must be equal.

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Now let's fill in the proof table:

| STATEMENTS | REASONS |
|-----------|--------|
| 1. $ m\angle 1 = m\angle 3 $ | 1. Given |
| 2. $ m\angle EBA = m\angle 3 + m\angle 2 $ | 2. Angle Addition Postulate |
| 3. $ m\angle DBC = m\angle 1 + m\angle 2 $ | 3. Angle Addition Postulate |
| 4. $ m\angle EBA = m\angle 1 + m\angle 2 $ | 4. Substitution (from statement 1) |
| 5. $ m\angle EBA = m\angle DBC $ | 5. Transitive Property of Equality |

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Final Proof Table:

| STATEMENTS | REASONS |
|-----------|--------|
| 1. $ m\angle 1 = m\angle 3 $ | 1. Given |
| 2. $ m\angle EBA = m\angle 3 + m\angle 2 $ | 2. Angle Addition Postulate |
| 3. $ m\angle DBC = m\angle 1 + m\angle 2 $ | 3. Angle Addition Postulate |
| 4. $ m\angle EBA = m\angle 1 + m\angle 2 $ | 4. Substitution (since $ m\angle 3 = m\angle 1 $) |
| 5. $ m\angle EBA = m\angle DBC $ | 5. Transitive Property of Equality |

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Explanation of Each Step:

- Step 1: Given.
- Step 2: $ \angle EBA $ is made up of $ \angle 3 $ and $ \angle 2 $, so their measures add.
- Step 3: Similarly, $ \angle DBC $ is made up of $ \angle 1 $ and $ \angle 2 $.
- Step 4: Since $ m\angle 1 = m\angle 3 $, we substitute $ m\angle 3 $ with $ m\angle 1 $ in Step 2 → now both expressions are $ m\angle 1 + m\angle 2 $.
- Step 5: Since both angles equal the same expression, they are equal by the transitive property.

✔ Therefore, $ m\angle EBA = m\angle DBC $ is proven.