In the given diagram, we are asked to find the value of angle $A$ when angles $B = 56^\circ$ and $C = 75^\circ$.

This appears to be a problem involving angles in a triangle, where $A^\circ$, $B^\circ$, and $C^\circ$ represent the interior angles of a triangle. The key geometric principle to use here is that the sum of the interior angles in any triangle is always $180^\circ$.

That is: $A + B + C = 180^\circ$

Given: $B = 56^\circ, \quad C = 75^\circ$

Substitute these values into the equation: $A + 56^\circ + 75^\circ = 180^\circ$ $A + 131^\circ = 180^\circ$

Now, solve for $A$: $A = 180^\circ - 131^\circ = 49^\circ$

Thus, the value of $A$ is $49^\circ$.

Would you like further details, or do you have any questions?

Here are 5 related questions to extend this problem:

1. What happens to the sum of the angles if the triangle is non-Euclidean (on a curved surface)?
2. How would you calculate the exterior angle at $A$?
3. Can you prove that the sum of the interior angles of a triangle is always 180 degrees?
4. What if the angle $C$ were changed to $85^\circ$? How would that affect angle $A$?
5. How would you solve for $A$ if you were only given the exterior angles of the triangle?

Tip: When solving triangle problems, always check whether it's necessary to apply trigonometric rules or simply use the angle sum property.