https://file.aiquickdraw.com/mathbot/file/17259854511587oaz1htu.jpg

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.

Given: BC bisects ∠ABD and m∠ABD = 52°. Prove: m∠ABC = 26°.

Learn how to prove that m∠ABC = 26° when BC bisects ∠ABD and m∠ABD = 52°. This proof involves key geometry concepts, including the angle bisector definition, angle addition postulate, and substitution property. Suitable for students in Grades 8-10.

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