2026 AMC 10A Problem 4

First, a $190^\circ$ angle is allowed. It is bigger than a straight angle, so at that corner the polygon bends inward, like the notch in a star or an arrowhead. The usual angle-sum rule still holds for such a polygon.

1. Recall the angle sum. A polygon with $n$ sides has $n$ corners, and its inside angles add up to
$$180(n - 2) \text{ degrees}.$$
For example, a triangle ($n = 3$) gives $180$ and a square ($n = 4$) gives $360$.

2. Add up the given angles. There are $n$ angles: $\tfrac{n}{2}$ of them are $150^\circ$ and $\tfrac{n}{2}$ are $190^\circ$. Pair each $150^\circ$ angle with a $190^\circ$ angle. Each pair adds to $340^\circ$, and there are $\tfrac{n}{2}$ pairs, so the total is
$$\tfrac{n}{2} \times 340 = 170n.$$
(Another way to see it: the angles average $\tfrac{150 + 190}{2} = 170^\circ$, and there are $n$ of them.)

3. Set the two expressions equal.
$$170n = 180(n - 2) = 180n - 360.$$
Subtract $170n$ from both sides: $0 = 10n - 360$, so $n = 36$.

1. Check. With $36$ sides there are $18$ angles of each size: $18 \times 150 + 18 \times 190 = 2700 + 3420 = 6120$. And $180 \times (36 - 2) = 180 \times 34 = 6120$. They match.

The answer is $\boxed{\textbf{(A)}\ 36}$.

The angle sum of a polygon depends only on how many sides it has, not on its shape, even when some corners point inward. So any problem that tells you the sizes of the angles turns into one equation: "the angles I am given add up to $180(n-2)$." When the angles come in equal groups, replacing them by their average makes the equation short.

Walk around the polygon and track how much you turn.

1. At a $150^\circ$ corner you turn $180 - 150 = 30^\circ$ to follow the next side.
2. At a $190^\circ$ corner the polygon bends the other way, so you turn back by $190 - 180 = 10^\circ$. Count that as $-10^\circ$.
3. One corner of each kind together turns you $30 - 10 = 20^\circ$.
4. Going all the way around a polygon, you turn a full $360^\circ$ in total. So there are $360 \div 20 = 18$ pairs of corners, which is $36$ corners and $36$ sides.