To solve for the measure of $\angle 7$, we can use properties of parallel lines and transversals, since the diagram shows a set of intersecting lines, likely forming a parallelogram or a similar configuration.

Given Information:

• $m\angle 3 = 75^\circ$
• $m\angle 10 = 105^\circ$

Approach:

1. Identify corresponding and supplementary angles:

   • From the figure, angles on opposite sides of the same transversal are either corresponding angles or supplementary angles.
   • Since angles $\angle 3$ and $\angle 7$ are on parallel lines and intersected by a transversal, they are alternate interior angles.

2. Apply the alternate interior angle theorem:

   • The alternate interior angle theorem states that if two lines are parallel and are cut by a transversal, then the alternate interior angles are equal.

Therefore: $m\angle 7 = m\angle 3 = 75^\circ$

Final Answer:

$m\angle 7 = 75^\circ$

Would you like more details on the reasoning or explanation?

Here are 5 related questions to extend your understanding:

1. What is the relationship between corresponding angles in a set of parallel lines?
2. How would you find the measure of $\angle 9$ in the given diagram?
3. How do supplementary angles help in determining the measure of other angles in parallel line setups?
4. Can you identify all the pairs of alternate interior angles in this diagram?
5. What other angle properties can be used when transversals cut parallel lines?

Tip: Remember that alternate interior angles are equal when the lines are parallel!