1. \( \angle 1 \) and \( \angle 2 \) are vertical angles. If \( m \angle 1= \) \( 6 x+11 \) and \( m \angle 2=10 x-9 \), find \( m \angle 1 \).

Since \( \angle 1 \) and \( \angle 2 \) are vertical angles, they are equal. Therefore, we can set their measures equal to each other: \[ 6x + 11 = 10x - 9 \] To solve for \( x \), we first rearrange the equation by moving all terms involving \( x \) to one side and constant terms to the other side. We can do this by subtracting \( 6x \) from both sides: \[ 11 = 10x - 6x - 9 \] This simplifies to: \[ 11 = 4x - 9 \] Next, we add \( 9 \) to both sides: \[ 11 + 9 = 4x \] \[ 20 = 4x \] Now, we divide both sides by \( 4 \): \[ x = 5 \] Now that we have the value of \( x \), we can substitute it back into the expression for \( m \angle 1 \): \[ m \angle 1 = 6x + 11 \] Substituting \( x = 5 \): \[ m \angle 1 = 6(5) + 11 \] \[ m \angle 1 = 30 + 11 \] \[ m \angle 1 = 41 \] Thus, the measure of \( \angle 1 \) is \[ \boxed{41} \]