### Instruction: The consecutive angles of a particular trapezoid form an arithmetic sequence.  If the largest angle measures $120^{\circ}$, what is the measure of the smallest angle?### Response: I know that an arithmetic sequence is a sequence of numbers where each term is obtained by adding a constant amount to the previous term.
I also know that the sum of the interior angles of any quadrilateral is $360^{\circ}$.
So, let me call the smallest angle $x$, and the constant amount $d$.
Then, the other angles are $x + d$, $x + 2d$, and $x + 3d$.
The largest angle is $x + 3d$, so I can set that equal to $120^{\circ}$ and solve for $x$ and $d$.
I get $x + 3d = 120$.
I can also use the fact that the angles add up to $360^{\circ}$.
That gives me $x + (x + d) + (x + 2d) + (x + 3d) = 360$.
I can simplify this equation by combining like terms.
I get $4x + 6d = 360$.
Now I have two equations and two unknowns, so I can use substitution or elimination to solve for $x$ and $d$.
I think elimination might be easier, since I can multiply the first equation by $-2$ and add it to the second equation to eliminate $d$.
That gives me $-2x - 6d + 4x + 6d = -240 + 360$.
Simplifying further, I get $2x = 120$.
Dividing by $2$, I get $x = 60$.
So, the smallest angle is $60^{\circ}$.
I can also find $d$ by plugging $x = 60$ into the first equation.
That gives me $60 + 3d = 120$.
Subtracting $60$, I get $3d = 60$.
Dividing by $3$, I get $d = 20$.
So, the angles are $60^{\circ}$, $80^{\circ}$, $100^{\circ}$, and $120^{\circ}$.