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The problem involves a triangle with various angles labeled from \( m \angle 1 \) to \( m \angle 7 \). The given values are \( m \angle 3 = 105^\circ \) and \( m \angle 4 = 140^\circ \). The task is to find the measures of the other angles.

### Analyzing the Diagram:

1. **Linear Pair Relationships**:
   - \( m \angle 3 \) and \( m \angle 2 \) are on a straight line, so they form a linear pair. Therefore:
     \[
     m \angle 3 + m \angle 2 = 180^\circ
     \]
     Substituting \( m \angle 3 = 105^\circ \), we get:
     \[
     m \angle 2 = 180^\circ - 105^\circ = 75^\circ
     \]

   - Similarly, \( m \angle 4 \) and \( m \angle 1 \) form a linear pair:
     \[
     m \angle 4 + m \angle 1 = 180^\circ
     \]
     Substituting \( m \angle 4 = 140^\circ \), we get:
     \[
     m \angle 1 = 180^\circ - 140^\circ = 40^\circ
     \]

2. **Vertically Opposite Angles**:
   - Vertically opposite angles are equal, so:
     \[
     m \angle 2 = m \angle 5 = 75^\circ
     \]
     \[
     m \angle 1 = m \angle 6 = 40^\circ
     \]

3. **Sum of Angles on a Straight Line**:
   - For angles 3, 7, and 5, \( m \angle 7 \) is a part of the straight line with \( m \angle 3 \) and \( m \angle 5 \), so:
     \[
     m \angle 7 + m \angle 3 = 180^\circ
     \]
     Substituting \( m \angle 3 = 105^\circ \), we get:
     \[
     m \angle 7 = 180^\circ - 105^\circ = 75^\circ
     \]

### Summary of Angle Measures:
- \( m \angle 1 = 40^\circ \)
- \( m \angle 2 = 75^\circ \)
- \( m \angle 5 = 75^\circ \)
- \( m \angle 6 = 40^\circ \)
- \( m \angle 7 = 75^\circ \)

Would you like more details or have any further questions?

Here are five related questions to expand your understanding:
1. How do vertically opposite angles form?
2. What is the significance of linear pairs in geometry?
3. Can you prove that the sum of angles in a triangle is always \(180^\circ\)?
4. How do you determine if two angles are supplementary?
5. How would the calculations change if \( m \angle 4 \) were different?

**Tip:** Understanding the properties of vertically opposite angles and linear pairs is fundamental to solving many geometry problems involving angles.

If m∠3 = 105° and m∠4 = 140°, find the other angles in the given geometric figure.

Learn how to find the unknown angles in a geometric figure where m∠3 = 105° and m∠4 = 140°. This problem involves key geometry concepts such as linear pairs, vertically opposite angles, and the sum of angles on a straight line. Suitable for students in Grades 7-9.

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