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

The given angles are: $77^\circ, \ 56^\circ, \ \text{and } x^\circ.$

Using the formula: $77^\circ + 56^\circ + x = 180^\circ$

Simplify: $133^\circ + x = 180^\circ$

Solve for $x$: $x = 180^\circ - 133^\circ$ $x = 47^\circ$

Thus, the value of $x$ is $47^\circ$.

Let me know if you need further explanation or have questions. Here are some related problems to explore:

1. What happens to the angles if the triangle is a right triangle?
2. How do you find missing angles in polygons with more than three sides?
3. What is the sum of the angles of a quadrilateral, and how is it derived?
4. How do exterior angles relate to the interior angles of a triangle?
5. What is the difference between solving angles in acute vs. obtuse triangles?

Tip: Always double-check your calculations by ensuring the sum of all angles matches $180^\circ$ in a triangle.