In the image, we are given a triangle $\triangle ABC$ with the following angle measures:

• $m \angle A = 8x$
• $m \angle B = 10x$
• $m \angle C = 7x + 5$

Since the sum of the angles in a triangle is always $180^\circ$, we can set up the following equation:

$8x + 10x + (7x + 5) = 180$

Simplifying, we get:

$25x + 5 = 180$

Subtracting 5 from both sides:

$25x = 175$

Dividing both sides by 25:

$x = 7$

Now, substitute $x = 7$ back into the expression for $m \angle A$:

$m \angle A = 8x = 8 \times 7 = 56$

Thus, the measure of $\angle A$ is $56^\circ$.

Answer: The correct choice is 56.

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

Here are five related questions:

1. How can we verify if a triangle is valid using its angles?
2. What would be the measures of $\angle B$ and $\angle C$ in this triangle?
3. How would the solution change if the sum of angles in a polygon other than a triangle was used?
4. What are the conditions for solving unknowns in triangles with algebra?
5. Can you use trigonometric functions to determine side lengths given these angle measures?

Tip: Always check that the sum of calculated angles adds up to 180 degrees for a triangle, ensuring accuracy in your solution.