### Instruction: In trapezoid $ABCD$, sides $\overline{AB}$ and $\overline{CD}$ are parallel, $\angle A = 2\angle D$, and $\angle C = 3\angle B$.  Find $\angle B$.### Response: I notice that the trapezoid has two parallel sides, so the angles on the same side of the transversal are supplementary.
That means $\angle A + \angle D = 180^\circ$ and $\angle C + \angle B = 180^\circ$.
I also notice that the problem gives me a relationship between $\angle A$ and $\angle D$, and between $\angle C$ and $\angle B$.
I can use those relationships to write a system of equations in terms of $\angle D$ and $\angle B$.
I have $\angle A = 2\angle D$, so I can substitute $2\angle D$ for $\angle A$ in the first equation.
That gives me $2\angle D + \angle D = 180^\circ$, or $3\angle D = 180^\circ$.
I can solve for $\angle D$ by dividing both sides by $3$. I get $\angle D = 60^\circ$.
Now that I have $\angle D$, I can use the other relationship to find $\angle B$.
I have $\angle C = 3\angle B$, so I can substitute $3\angle B$ for $\angle C$ in the second equation.
That gives me $3\angle B + \angle B = 180^\circ$, or $4\angle B = 180^\circ$.