To solve for $x$ in this triangle, we will use the fact that the sum of the interior angles in a triangle is always $180^\circ$.

The angles given are:

1. $83^\circ$
2. $61^\circ$
3. $(4x + 16)^\circ$

Setting up the equation: $83 + 61 + (4x + 16) = 180$

Let's simplify and solve for $x$:

1. Combine the constants on the left side: $144 + 4x + 16 = 180$

2. Simplify further: $160 + 4x = 180$

3. Subtract 160 from both sides: $4x = 20$

4. Divide by 4: $x = 5$

Thus, $x = 5$.

Would you like further details or have any questions?

Here are some related questions you might explore:

1. What is the measure of the third angle in the triangle?
2. How do we verify that the solution satisfies the conditions of a triangle?
3. How would the solution change if one angle was obtuse?
4. What is the importance of the triangle angle sum property in geometry?
5. How can we apply similar techniques to find unknown sides using trigonometry?

Tip: When solving for unknowns in a triangle, always ensure the sum of the angles equals $180^\circ$ as a quick verification step.