The diagram described involves two intersecting lines, $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$, which are perpendicular at point $X$. The ray $\overrightarrow{XY}$ divides the angle $\angle AXB$ into two smaller angles: $\angle AXY$ (labeled as $m$) and $\angle YXB$ (labeled as $n$).

Given this setup, the relationship between $\blueD{\angle m}$ (i.e., $\angle AXY$) and $\greenD{\angle n}$ (i.e., $\angle YXB$) is as follows:

Since $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$ are perpendicular, the full angle $\angle AXB$ is a right angle, meaning: $\angle AXB = 90^\circ.$

The ray $\overrightarrow{XY}$ splits $\angle AXB$ into the two angles $\angle AXY$ and $\angle YXB$, giving the equation: $\blueD{\angle m} + \greenD{\angle n} = 90^\circ.$ Therefore, the relationship between $\blueD{\angle m}$ and $\greenD{\angle n}$ is that they are complementary angles: $\blueD{\angle m} + \greenD{\angle n} = 90^\circ.$

Would you like further details or clarification?

Here are five related questions for further exploration:

1. How can you calculate $\angle m$ if $\angle n$ is known?
2. What are complementary angles, and why is their sum $90^\circ$?
3. How can you construct perpendicular lines geometrically?
4. Can you extend this concept to obtuse or acute angles?
5. What is the significance of the point $X$ in this configuration?

Tip: Complementary angles always sum up to 90 degrees, a key property when working with right-angle triangles and perpendicular lines!