In trapezoid $ABCD$, sides $\overline{AB}$ and $\overline{CD}$ are parallel, $\angle A = 2\angle D$, and $\angle C = 3\angle B$.  Find $\angle A$.
Since $\overline{AB}\parallel\overline{CD}$, we have $\angle A+ \angle D = 180^\circ$.  Since $\angle A = 2\angle D$, we have $2\angle D + \angle D = 180^\circ$, so $3\angle D = 180^\circ$, which means $\angle D = 60^\circ$.  Therefore, $\angle A = 2\angle D = \boxed{120^\circ}$.

[asy]
pair A,B,C,D;

A = (0,0);
B = (1,0);
D = rotate(120)*(0.8,0);
C = intersectionpoint(D--(D + (40,0)), B--(B + (rotate(135)*(1,0))));
draw(A--B--C--D--A);
label("$A$",A,SW);
label("$B$", B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
[/asy]
120
Level 5
Prealgebra
Problem: In trapezoid $ABCD$, sides $\overline{AB}$ and $\overline{CD}$ are parallel, $\angle A = 2\angle D$, and $\angle C = 3\angle B$.  Find $\angle A$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>In trapezoid $ABCD$, since $\overline{AB}$ and $\overline{CD}$ are parallel, the consecutive angles formed by the transversal lines are supplementary. Therefore, we have the relationships: 

1. $\angle A + \angle D = 180^\circ$ (1)
2. $\angle B + \angle C = 180^\circ$ (2)

Given that $\angle A = 2\angle D$, we can express $\angle D$ in terms of $\angle A$:

$\angle D = \frac{1}{2} \angle A$.

Substituting this into equation (1):

$\angle A + \frac{1}{2} \angle A = 180^\circ$.

This simplifies to:

$\frac{3}{2} \angle A = 180^\circ$.

Multiplying both sides by $\frac{2}{3}$ gives:

$\angle A = 120^\circ$.

Next, we also know that $\angle C = 3\angle B$. Substituting $\angle C$ into equation (2):

$\angle B + 3\angle B = 180^\circ$.

This simplifies to:

$4\angle B = 180^\circ$.

Dividing both sides by 4 yields:

$\angle B = 45^\circ$.

Now substituting $\angle B$ back to find $\angle C$:

$\angle C = 3 \times 45^\circ = 135^\circ$.

Finally, we verify the angle relationships:

- For angles at $A$ and $D$: $\angle A + \angle D = 120^\circ + 60^\circ = 180^\circ$.
- For angles at $B$ and $C$: $\angle B + \angle C = 45^\circ + 135^\circ = 180^\circ$.

All conditions are satisfied, confirming that the calculations are consistent. Thus, the final answer is $120$.<|END_OF_LATENT|> The final answer is $120$.