Given two angles (4m - 15) and (3m + 75) are vertically opposite, find the value of m.

The two angles, $4m - 15$ and $3m + 75$, are vertically opposite angles. Vertically opposite angles are equal.

Subtract $3m$ from both sides:

Add 15 to both sides:

The value of $m$ is 90.

If $m = 90$:
First angle: $4(90) - 15 = 360 - 15 = 345^\circ$
Second angle: $3(90) + 75 = 270 + 75 = 345^\circ$
The angles are equal, so the value of $m$ is correct.

The value of $m$ is $\boxed{\text{90}}$.