The image shows a geometry proof involving the bisecting of an angle. Let’s break it down:

Given:

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

To Prove:

• $m \angle ABC = 26^\circ$.

Steps to the Proof:

1. $m \angle ABD = 52^\circ$ — (Reason: Given)
2. $BC$ bisects $\angle ABD$ — (Reason: Given)
3. $m \angle ABC = m \angle CBD$ — (Reason: Definition of bisector)
4. $m \angle ABC + m \angle CBD = m \angle ABD$ — (Reason: Angle addition postulate)
5. $m \angle ABC + m \angle ABC = 52^\circ$ — (Reason: Substitution property)

Now, for the final statement $m \angle ABC + m \angle ABC = 52^\circ$, the correct reason to select would be substitution property, as it directly follows from substituting $m \angle CBD$ with $m \angle ABC$ due to the bisector property.

Conclusion:

Since the angle was bisected, we divide the total angle into two equal parts. So:

$2m \angle ABC = 52^\circ \implies m \angle ABC = \frac{52^\circ}{2} = 26^\circ.$

Would you like more details or have any questions? Here are five related questions to expand on this:

1. How do you apply the angle addition postulate in more complex geometrical figures?
2. Why is the definition of a bisector important in triangle congruence proofs?
3. Can the substitution property be used for algebraic expressions involving angles?
4. How can we prove that two angles are equal if a bisector is not given?
5. What are other common properties involving bisectors in triangles?

Tip:

Always check if you can simplify the angles by looking for symmetries or properties like bisectors to make your proof easier.