Question

m∠3 = 82°, find m∠4.
m∠4 = 52°

Explanation:

Step1: Vertical angles

$\angle2$ and $\angle4$ are vertical angles. Vertical angles are equal. But wait, no, let's check again. Wait, $\angle3$ and $\angle1$ are vertical angles. $\angle1+\angle2 = 180^{\circ}$ (linear - pair), $\angle1+\angle4=180^{\circ}$ (linear - pair). So $\angle2=\angle4$? No, wrong. Wait, $\angle3$ and $\angle1$ are vertical angles. $\angle1+\angle2 = 180^{\circ}$, $\angle3+\angle4 = 180^{\circ}$. Since $\angle2 = 82^{\circ}$, and $\angle3$ and $\angle2$ are vertical angles? No. Wait, $\angle3$ and $\angle1$ are vertical angles. $\angle1+\angle2=180^{\circ}\Rightarrow\angle1 = 180^{\circ}-\angle2=180 - 82=98^{\circ}$. Then $\angle3=\angle1 = 98^{\circ}$ (vertical angles). But wait, no, another approach: $\angle3$ and $\angle2$ are adjacent angles forming a linear pair? No. Wait, $\angle3$ and $\angle4$ form a linear pair. $\angle3+\angle4=180^{\circ}$.

Step2: Solve for $\angle4$

Given $\angle3 = 82^{\circ}$, using the linear - pair relationship $\angle3+\angle4 = 180^{\circ}$. Then $\angle4=180^{\circ}-\angle3$. Substitute $\angle3 = 82^{\circ}$ into the formula: $\angle4=180 - 82=98^{\circ}$.

Answer:

$98$