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.