Let's analyze the problem.

The figure presents an angle relationship involving three angles at point $N$, namely:

• $\angle KNM = 31^\circ$
• $\angle KJN = 74^\circ$ (labeled in red)
• $\angle MKN = (2x - 5)^\circ$

These angles form a linear pair since $K, N,$ and $J$ are along a straight line. Therefore, the sum of the angles around $N$ on a straight line must equal $180^\circ$.

We can form the equation: $\angle KJN + \angle KNM + \angle MKN = 180^\circ$ Substitute the known values: $74^\circ + 31^\circ + (2x - 5)^\circ = 180^\circ$ Now, let's solve for $x$. I'll calculate that.The value of $x$ is $40$.

Now, to find $y$, we can substitute $x = 40$ into the expression for the angle $MKN$, which is $2x - 5$. I'll calculate that next.The angle $MKN$ is $75^\circ$, so $y = 75$.

To summarize:

• $x = 40$
• $y = 75$

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the definition of a linear pair of angles?
2. How do you calculate unknown angles using algebraic expressions?
3. Can supplementary angles always add up to $180^\circ$?
4. How do you check if an angle equation is set up correctly in geometry problems?
5. What are the different types of angle relationships found in geometry?

Tip: Always double-check if the angles in your problem form complementary, supplementary, or vertical pairs, as these relationships often lead to solving unknowns easily!