\( 2 m \angle x+m \angle z=180^{\circ} \) These angles are \( \square \) , so their measures sum to \( 180^{\circ} \). 3 \( m \angle y+m \angle z=180^{\circ} \) Same as the previous reason. \( 4 \quad m \angle x+m \angle z= \) Substitution (2, 3). \( 5 \quad m \angle x=m \angle y \) Subtract \( m \angle z \) from both sides of the equation (4).

Let's analyze the problem step by step, focusing on the relationships between the angles and the equations provided. ### Known Conditions 1. \( 2 m \angle x + m \angle z = 180^{\circ} \) (Equation 1) 2. \( 3 m \angle y + m \angle z = 180^{\circ} \) (Equation 2) 3. We need to find \( m \angle x \) in terms of \( m \angle y \). ### Step 1: Solve for \( m \angle z \) in terms of \( m \angle x \) and \( m \angle y \) From Equation 1: \[ m \angle z = 180^{\circ} - 2 m \angle x \] From Equation 2: \[ m \angle z = 180^{\circ} - 3 m \angle y \] ### Step 2: Set the two expressions for \( m \angle z \) equal to each other Equating the two expressions for \( m \angle z \): \[ 180^{\circ} - 2 m \angle x = 180^{\circ} - 3 m \angle y \] ### Step 3: Simplify the equation Subtract \( 180^{\circ} \) from both sides: \[ -2 m \angle x = -3 m \angle y \] ### Step 4: Rearranging the equation Multiply through by -1: \[ 2 m \angle x = 3 m \angle y \] ### Step 5: Solve for \( m \angle x \) Now, divide both sides by 2: \[ m \angle x = \frac{3}{2} m \angle y \] ### Conclusion Thus, we have shown that: \[ m \angle x = \frac{3}{2} m \angle y \] This means that the measure of angle \( x \) is \( \frac{3}{2} \) times the measure of angle \( y \).