The image shows a triangle with three angle measures expressed in terms of $x$:

• $(2x + 6)^\circ$
• $(5x + 1)^\circ$
• $(2x - 7)^\circ$

Since the sum of the interior angles of a triangle is always $180^\circ$, we can write the equation as follows:

$(2x + 6) + (5x + 1) + (2x - 7) = 180$

Now, let's simplify the left-hand side:

$2x + 5x + 2x + 6 + 1 - 7 = 180$

$9x = 180$

Now, solve for $x$:

$x = \frac{180}{9} = 20$

So, $x = 20$.

Verifying the solution:

Now substitute $x = 20$ into the expressions for the angles:

• First angle: $2(20) + 6 = 40 + 6 = 46^\circ$
• Second angle: $5(20) + 1 = 100 + 1 = 101^\circ$
• Third angle: $2(20) - 7 = 40 - 7 = 33^\circ$

Finally, check the sum: $46^\circ + 101^\circ + 33^\circ = 180^\circ$

Since the angles sum to 180°, the solution is correct.

Would you like more details or have any questions? Here are some related problems for further practice:

1. What are the general properties of the interior angles of any polygon?
2. How would you solve for $x$ if the angles were exterior angles instead?
3. If one of the angles was missing, how could you solve for it given the other two?
4. How can you prove the angle sum property of a triangle using geometry?
5. What happens to the angles of a triangle if the triangle is isosceles or equilateral?

Tip: Always check your work by verifying that the sum of the angles meets the known geometric property—like 180° for a triangle!