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We are given a triangle with three angles, one of which is expressed in terms of \(x\). The angle measures are:

- \(106^\circ\)
- \(31^\circ\)
- \( (2x + 1)^\circ \)

Since the sum of the interior angles of a triangle is always \(180^\circ\), we can set up the following equation to solve for \(x\):

\[
106^\circ + 31^\circ + (2x + 1)^\circ = 180^\circ
\]

First, simplify the left-hand side:

\[
137^\circ + (2x + 1)^\circ = 180^\circ
\]

Next, combine constants:

\[
138^\circ + 2x = 180^\circ
\]

Now, subtract \(138^\circ\) from both sides:

\[
2x = 180^\circ - 138^\circ
\]

\[
2x = 42^\circ
\]

Finally, divide by 2:

\[
x = 21^\circ
\]

Thus, the value of \(x\) is \(21^\circ\).

Let me know if you'd like further details or have any questions!

Here are 5 related questions you might find useful:
1. What is the sum of the angles in any polygon?
2. How can you verify the solution for \(x\) by calculating all angles in the triangle?
3. What happens if two angles of a triangle are known but the third is given in terms of \(x\)?
4. How do angle relationships change in non-Euclidean geometry?
5. How would this problem change if the triangle were non-acute or obtuse?

**Tip:** In any triangle, always remember that the sum of the three angles will add up to \(180^\circ\).

The measures of the angles of a triangle are shown in the figure below. Solve for x.

Learn how to solve for x in a triangle where the angles are 106°, 31°, and (2x + 1)°. This problem involves algebraic manipulation and the triangle angle sum theorem, suitable for students in Grades 7-9.

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