To solve for the angle $x$ in this pentagon, we can use the fact that the sum of the interior angles of a polygon with $n$ sides is given by:

$\text{Sum of interior angles} = (n - 2) \times 180^\circ$

Since this is a pentagon ($n = 5$), we calculate the sum of its interior angles as:

$(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$

Now, we add up the known angles and set up an equation to solve for $x$.

The known angles are:

• $57^\circ$
• $27^\circ$
• $131^\circ$
• $61^\circ$

Adding these:

$57^\circ + 27^\circ + 131^\circ + 61^\circ = 276^\circ$

Since the sum of all interior angles is $540^\circ$, we can find $x$ by subtracting the sum of the known angles from $540^\circ$:

$x = 540^\circ - 276^\circ = 264^\circ$

Therefore, the value of $x$ is $264^\circ$.

If you have further questions or need clarification, feel free to ask!

Related Questions to Explore Further:

1. How do you find the sum of interior angles for any polygon?
2. Why does a pentagon have a total of $540^\circ$ in its interior angles?
3. What would be the sum of the exterior angles of any polygon, and why?
4. How can you verify the calculation for the angle $x$ using exterior angles?
5. How would the approach change if it were a regular pentagon (all angles and sides equal)?

Tip:

For any polygon, the sum of the exterior angles is always $360^\circ$, regardless of the number of sides. This is a useful property when solving problems involving polygons.