To help complete this two-column proof, let's break down the key information and work through the logic:

Given:

• $BC$ bisects $\angle ABD$
• $m\angle ABD = 52^\circ$

To Prove:

• $m\angle ABC = 26^\circ$

Definitions and Properties:

• Since $BC$ bisects $\angle ABD$, it divides $\angle ABD$ into two equal angles: $\angle ABC$ and $\angle CBD$.
• Therefore, $m\angle ABC = m\angle CBD$.

We can use this information to fill out the two-column proof.

Solution:

Statements:

1. $m\angle ABD = 52^\circ$
2. $BC$ bisects $\angle ABD$
3. $m\angle ABC = m\angle CBD$
4. $m\angle ABC + m\angle CBD = m\angle ABD$
5. $m\angle ABC + m\angle ABC = 52^\circ$
6. $2 \times m\angle ABC = 52^\circ$
7. $m\angle ABC = 26^\circ$

Reasons:

1. Given
2. Given
3. Definition of an angle bisector
4. Angle addition postulate
5. Substitution
6. Simplification
7. Division property of equality

With these steps, the proof is completed, showing that $m\angle ABC = 26^\circ$.

Would you like more details or have any questions? Here are some follow-up questions to deepen your understanding:

1. What is the definition of an angle bisector?
2. How does the angle addition postulate apply to this situation?
3. What are some other situations where angle bisectors are useful in geometry?
4. How would the proof change if $BC$ did not bisect $\angle ABD$?
5. Can you explain why we divide by 2 in step 6?

Tip: Always ensure that you list all given information clearly in the proof to maintain logical flow!