We are given the measures of the angles of triangle $ABC$ in terms of $x$:

• $m\angle A = \left( \frac{8}{3}x - 5 \right)^\circ$,
• $m\angle B = (2x + 25)^\circ$,
• $m\angle C = 2\left( x + \frac{5}{2} \right)^\circ$.

The sum of the angles in a triangle is always $180^\circ$. Therefore:

$m\angle A + m\angle B + m\angle C = 180^\circ.$

Step 1: Substitute the given expressions

$\left( \frac{8}{3}x - 5 \right) + (2x + 25) + 2\left( x + \frac{5}{2} \right) = 180.$

Step 2: Simplify each term

• $\left( \frac{8}{3}x - 5 \right)$ remains as is.
• $2(x + \frac{5}{2}) = 2x + 5$.

Substituting into the equation:

$\frac{8}{3}x - 5 + 2x + 25 + 2x + 5 = 180.$

Step 3: Combine like terms

Combine all the $x$-terms and constants:

• $\frac{8}{3}x + 2x + 2x = \frac{8}{3}x + \frac{6}{3}x + \frac{6}{3}x = \frac{20}{3}x$,
• $-5 + 25 + 5 = 25$.

So the equation becomes:

$\frac{20}{3}x + 25 = 180.$

Step 4: Solve for $x$

First, subtract 25 from both sides:

$\frac{20}{3}x = 155.$

Multiply through by 3 to eliminate the fraction:

$20x = 465.$

Divide by 20:

$x = \frac{465}{20} = 23.25.$

Step 5: Find $m\angle B$

Substitute $x = 23.25$ into $m\angle B = 2x + 25$:

$m\angle B = 2(23.25) + 25.$

Simplify:

$m\angle B = 46.5 + 25 = 71.5.$