In the given image, you are asked to find the measure of $m \angle 5$. The diagram shows that $m \angle 5 = m \angle 6$, and these angles form a linear pair with a right angle (90°), indicated by the square symbol between them.

Since $\angle 5$ and $\angle 6$ are congruent (equal), and together they form a straight line (180°), we can set up an equation:

$m \angle 5 + m \angle 6 = 90^\circ$

But since $m \angle 5 = m \angle 6$, we can substitute:

$2m \angle 5 = 90^\circ$

Solving for $m \angle 5$:

$m \angle 5 = \frac{90^\circ}{2} = 45^\circ$

Thus, $m \angle 5 = 45^\circ$.

Would you like more details or have any further questions? Here are five related questions for further exploration:

1. What is a linear pair of angles, and how do they relate to the concept of supplementary angles?
2. Why are vertical angles always congruent?
3. How can you apply the concept of complementary angles in geometry problems?
4. What are alternate interior angles, and how do they relate to parallel lines?
5. How can you identify congruent angles in various geometric figures?

Tip: Always look for angle relationships such as complementary, supplementary, or vertical angles to solve for unknown measures in geometry problems.