The problem asks to find the value of $m$ given that the angles $4m - 15$ and $m + 75$ are vertical angles formed by two intersecting lines.

1. Problem Description

The problem asks to find the value of $m$ given that the angles $4m - 15$ and $m + 75$ are vertical angles formed by two intersecting lines.

2. Solution Steps

Vertical angles are equal. Therefore, we have the equation:

$4m - 15 = m + 75$

Subtract $m$ from both sides:

$4m - m - 15 = m - m + 75$

$3m - 15 = 75$

Add 15 to both sides:

$3m - 15 + 15 = 75 + 15$

$3m = 90$

Divide both sides by 3:

$\frac{3m}{3} = \frac{90}{3}$

$m = 30$

3. Final Answer

The value of $m$ is 30.