$$ 2 m \angle x+m \angle z=180^{\circ} $$ 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).

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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 $$ y $$.
$$ m \angle x = \frac{3}{2} m \angle y $$

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