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In the given figure, lines \( m \) and \( n \) are parallel, and the transversal \( t \) cuts through both lines. You are given that \( m\angle 4 = 116^\circ \). We can use properties of parallel lines and transversals to find the measures of the other angles.

### Key angle relationships:
1. **Corresponding angles**: Angles on the same side of the transversal and in the same relative position are equal.
2. **Alternate interior angles**: Angles on opposite sides of the transversal but between the parallel lines are equal.
3. **Linear pairs**: Adjacent angles on a straight line sum up to \( 180^\circ \).

### Now, let's solve for each angle:
- **\( m\angle 4 = 116^\circ \)** (Given)
- **\( m\angle 5 \)**: \( \angle 4 \) and \( \angle 5 \) form a linear pair, so they must sum to \( 180^\circ \). 
  \[
  m\angle 5 = 180^\circ - 116^\circ = 64^\circ
  \]
- **\( m\angle 3 \)**: \( \angle 3 \) and \( \angle 4 \) are vertical angles, so they are equal. 
  \[
  m\angle 3 = 116^\circ
  \]
- **\( m\angle 6 \)**: \( \angle 6 \) corresponds to \( \angle 3 \), so they are equal.
  \[
  m\angle 6 = 116^\circ
  \]
- **\( m\angle 7 \)**: \( \angle 7 \) corresponds to \( \angle 4 \), so they are equal.
  \[
  m\angle 7 = 116^\circ
  \]
- **\( m\angle 1 \)**: \( \angle 1 \) and \( \angle 8 \) form a linear pair, so:
  \[
  m\angle 1 = 180^\circ - 116^\circ = 64^\circ
  \]
- **\( m\angle 8 \)**: \( \angle 8 \) corresponds to \( \angle 5 \), so they are equal.
  \[
  m\angle 8 = 64^\circ
  \]
- **\( m\angle 2 \)**: \( \angle 2 \) corresponds to \( \angle 1 \), so they are equal.
  \[
  m\angle 2 = 64^\circ
  \]

### Summary of angle measures:
- \( m\angle 1 = 64^\circ \)
- \( m\angle 2 = 64^\circ \)
- \( m\angle 3 = 116^\circ \)
- \( m\angle 4 = 116^\circ \) (Given)
- \( m\angle 5 = 64^\circ \)
- \( m\angle 6 = 116^\circ \)
- \( m\angle 7 = 116^\circ \)
- \( m\angle 8 = 64^\circ \)

Would you like further clarification or details?

### 5 Related Questions:
1. What is the definition of alternate interior angles, and why are they equal in this context?
2. How are corresponding angles identified when a transversal crosses two parallel lines?
3. Why do adjacent angles on a straight line sum to \(180^\circ\)?
4. Can you explain what vertical angles are and why they are always equal?
5. How would the angle measures change if lines \(m\) and \(n\) were not parallel?

### Tip:
Always remember that in the case of parallel lines and a transversal, corresponding, alternate interior, and alternate exterior angles are always congruent!

In the figure, m is parallel to n and m∠4 = 116°. Find the measures of the other angles.

Explore how to find the measures of angles formed by parallel lines and a transversal when one angle is given as 116°. Learn about corresponding angles, alternate interior angles, and linear pairs, perfect for students in Grades 7-10.

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