1. **State the problem:** We are given expressions for angles 1, 2, and 4 in terms of variables $x$ and $y$: $$m\angle 1 = 4x + 3y + 6$$ $$m\angle 2 = 10x + 6y$$ $$m\angle 4 = 12x + 4y + 8$$ We need to find $m\angle 3$. 2. **Understand the geometry:** Angles 1, 2, 3, and 4 are formed by two intersecting lines. Opposite angles (vertical angles) are equal. Also, adjacent angles on a straight line sum to $180^\circ$. 3. **Use vertical angles:** Since angles 1 and 3 are vertical angles, they are equal: $$m\angle 3 = m\angle 1 = 4x + 3y + 6$$ 4. **Use linear pair:** Angles 1 and 2 are adjacent on a straight line, so: $$m\angle 1 + m\angle 2 = 180$$ Substitute expressions: $$4x + 3y + 6 + 10x + 6y = 180$$ Simplify: $$14x + 9y + 6 = 180$$ Subtract 6: $$14x + 9y = 174$$ 5. **Use linear pair for angles 3 and 4:** Angles 3 and 4 are adjacent on the other line, so: $$m\angle 3 + m\angle 4 = 180$$ Substitute $m\angle 3 = 4x + 3y + 6$ and $m\angle 4 = 12x + 4y + 8$: $$4x + 3y + 6 + 12x + 4y + 8 = 180$$ Simplify: $$16x + 7y + 14 = 180$$ Subtract 14: $$16x + 7y = 166$$ 6. **Solve the system:** $$\begin{cases} 14x + 9y = 174 \\ 16x + 7y = 166 \end{cases}$$ Multiply first equation by 7 and second by 9 to eliminate $y$: $$\begin{cases} 98x + 63y = 1218 \\ 144x + 63y = 1494 \end{cases}$$ Subtract first from second: $$144x - 98x + 63y - 63y = 1494 - 1218$$ $$46x = 276$$ Divide both sides by 46: $$x = \frac{\cancel{46}x}{\cancel{46}} = \frac{276}{46} = 6$$ 7. **Find $y$:** Substitute $x=6$ into first equation: $$14(6) + 9y = 174$$ $$84 + 9y = 174$$ Subtract 84: $$9y = 90$$ Divide both sides by 9: $$y = \frac{\cancel{9}y}{\cancel{9}} = \frac{90}{9} = 10$$ 8. **Find $m\angle 3$:** $$m\angle 3 = 4x + 3y + 6 = 4(6) + 3(10) + 6 = 24 + 30 + 6 = 60$$ **Final answer:** $$\boxed{m\angle 3 = 60}$$