https://file.aiquickdraw.com/mathbot/file/1726667365295mfjbp2n1.jpg

Let's analyze the questions based on the diagram.

### Problem 26:
- You're given that \( m\angle 1 = 126^\circ \), and you need to find \( m\angle 2 \).
- Angles 1 and 2 form a linear pair (they are adjacent and on a straight line). Linear pairs are supplementary, meaning they add up to \(180^\circ\).
  
  So,
  \[
  m\angle 1 + m\angle 2 = 180^\circ
  \]
  \[
  126^\circ + m\angle 2 = 180^\circ
  \]
  Solving for \( m\angle 2 \),
  \[
  m\angle 2 = 180^\circ - 126^\circ = 54^\circ
  \]

### Problem 27:
- You're given \( m\angle 3 = 27^\circ \) and \( m\angle 4 = 85^\circ \), and you need to find \( m\angle 2 \).
- Angles 2 and 3 are vertically opposite angles, and vertically opposite angles are always equal.

  Therefore, 
  \[
  m\angle 2 = m\angle 3 = 27^\circ
  \]

Would you like more details or have any questions on these solutions?

### 5 Related Questions:
1. What is the definition of vertically opposite angles?
2. How can we prove that linear pairs are supplementary?
3. What other types of angle pairs exist (e.g., complementary, corresponding)?
4. How would the result change if one of the angles were unknown?
5. Can you describe a scenario where vertically opposite angles wouldn't apply?

### Tip:
When working with linear pairs, always check if the angles are supplementary—this will help in solving many geometry problems!

26. If m∠1 = 126°, then m∠2 = _________.
27. If m∠3 = 27° and m∠4 = 85°, then m∠2 = _________.

Find the value of m∠2 using linear pair and vertically opposite angle properties. In problem 26, m∠1 = 126°, so m∠2 is calculated using the supplementary angle theorem. In problem 27, m∠3 = 27° and m∠4 = 85°, leading to m∠2 based on vertically opposite angles.

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