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Let's solve the problem.

We are given:
- \( m\angle ABC = 110^\circ \)
- \( m\angle 2 = 36^\circ \)

We need to find \( m\angle ABD \).

### Step-by-step solution:
1. \( \angle ABC \) is the entire angle from ray \( AB \) to ray \( BC \).
2. \( \angle ABD \) and \( \angle 2 \) (which is \( \angle DBC \)) are **adjacent angles** that together form \( \angle ABC \). Hence, their measures add up to the total measure of \( \angle ABC \):
   \[
   m\angle ABC = m\angle ABD + m\angle 2
   \]
   Substituting the known values:
   \[
   110^\circ = m\angle ABD + 36^\circ
   \]

3. Solve for \( m\angle ABD \):
   \[
   m\angle ABD = 110^\circ - 36^\circ = 74^\circ
   \]

### Final Answer:
\[
m\angle ABD = 74^\circ
\]

Would you like me to explain any steps in more detail or provide additional examples?

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Here are five follow-up questions to deepen understanding:
1. What are adjacent angles, and how do they relate to this problem?
2. How can you determine if two angles are complementary or supplementary?
3. If another angle, \( \angle EBD \), were added, how would you find its measure?
4. Can angles in the same configuration sum to more than \(180^\circ\)? Why or why not?
5. How do you distinguish between interior and exterior angles?

**Tip:** Always double-check angle relations (like adjacent or supplementary) to ensure your equations are correct!

Find the measure of ∠ABD if m∠ABC = 110° and m∠2 = 36°.

Discover how to find the measure of angle ABD when given m∠ABC = 110° and m∠2 = 36°. This problem covers essential geometry concepts involving adjacent angles, suitable for students in Grades 7-9.

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