To solve for $m \angle A$, let's analyze the triangle $\triangle ABC$ where:

1. $m \angle A = (5x - 27)^\circ$
2. $m \angle B = (7x - 26)^\circ$
3. $m \angle C = (10x - 23)^\circ$

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

$(5x - 27) + (7x - 26) + (10x - 23) = 180$

Step 1: Simplify the Equation

Combine the terms:

$5x + 7x + 10x - 27 - 26 - 23 = 180$

$22x - 76 = 180$

Step 2: Solve for $x$

Add 76 to both sides:

$22x = 256$

Now, divide by 22:

$x = \frac{256}{22} = 11.6364$

Step 3: Find $m \angle A$

Now that we have $x \approx 11.6364$, substitute this value back into $m \angle A = (5x - 27)$:

$m \angle A = 5(11.6364) - 27$

$m \angle A \approx 58.182 - 27 = 31.182^\circ$

Final Answer

$m \angle A \approx 31.18^\circ$

Would you like more details or have any questions?

Here are some related questions to explore further:

1. What is the measure of $m \angle B$ in this triangle?
2. What is the measure of $m \angle C$ in this triangle?
3. How would the solution change if the sum of angles were greater or less than $180^\circ$?
4. How would you set up similar problems for quadrilaterals where the sum of angles is $360^\circ$?
5. How can we verify that our solution for $x$ satisfies all angle measures in the triangle?

Tip: Always double-check calculations by plugging values back into the original equation to ensure they satisfy the conditions of the problem.