To solve for $x$ in this triangle problem, we can apply the fact that the sum of the angles in any triangle must be $180^\circ$.

The angles given are:

• $(x + 6)^\circ$ at point $O$
• $(3x - 16)^\circ$ at point $M$
• $x^\circ$ at point $N$

We can set up the following equation for the sum of the angles: $(x + 6) + (3x - 16) + x = 180$

Step 1: Simplify the equation.

$x + 6 + 3x - 16 + x = 180$ $5x - 10 = 180$

Step 2: Solve for $x$.

$5x = 190$ $x = 38$

Thus, the value of $x$ is $38^\circ$.

Would you like further details or clarifications?

Follow-up Questions:

1. How would the problem change if the angle at $O$ were doubled?
2. What if the angle at $M$ was decreased by $10^\circ$?
3. How do we determine the largest angle in this triangle?
4. How would you solve if this was an isosceles triangle?
5. What are the properties of the angles in an equilateral triangle?

Tip: Remember, the sum of the angles in any triangle is always $180^\circ$.