Angles & Polygons

What you need to know

• Interior angles. The angles of a polygon with $n$ sides add up to $(n-2)\cdot180^\circ$, because the diagonals from one corner cut it into $n-2$ triangles. Example: a hexagon's angles add up to $4\cdot180^\circ = 720^\circ$.
• Exterior angles. The exterior angles of any convex polygon (one at each corner) add up to $360^\circ$. So a regular $n$-gon has exterior angles of $\frac{360^\circ}{n}$ and interior angles of $180^\circ - \frac{360^\circ}{n}$. Example: a regular octagon has $45^\circ$ exterior and $135^\circ$ interior angles.
• Parallel lines. When a line crosses two parallel lines, alternate interior angles (a "Z" shape) are equal, and co-interior angles (a "C" shape) add up to $180^\circ$. Example: a $70^\circ$ angle has a $70^\circ$ Z-partner and a $110^\circ$ C-partner.
• Exterior angle of a triangle. It equals the sum of the two inside angles at the other two corners. Example: angles of $50^\circ$ and $65^\circ$ give a $115^\circ$ exterior angle at the third corner.
• Isosceles triangles. Their base angles are equal, and a regular polygon's equal sides hide many of them. Example: in regular pentagon $ABCDE$, $AB = BC$ and $\angle B = 108^\circ$, so $\angle BAC = \frac{180^\circ-108^\circ}{2} = 36^\circ$.
• Angles in a circle. A central angle (corner at the center) equals its arc. An inscribed angle (corner on the circle) is half its arc. Example: a $100^\circ$ arc has a $100^\circ$ central angle and a $50^\circ$ inscribed angle.
• Diagonals. A convex polygon with $n$ sides has $\frac{n(n-3)}{2}$ diagonals: each corner joins $n-3$ others (not itself or its neighbors), counting each diagonal twice. Example: an octagon has $\frac{8\cdot5}{2} = 20$.

How AMC 10 tests it

• Problems 1–8: one missing angle in a figure of parallel lines, a triangle, or a kite.
• "A regular polygon's interior angle is $x$ (or $k$ times its exterior angle); how many sides?" Solve $\frac{360^\circ}{n} =$ (exterior angle) for $n$. Example: $140^\circ$ inside means $40^\circ$ outside, so $n = 9$.
• Problems 10–16: a regular pentagon, hexagon, or octagon with diagonals, or with an equilateral triangle or square attached. The trick is spotting the isosceles triangles.

Standard approaches

1. Name unknown angles with letters, and write equations from triangle and polygon angle sums.
2. For regular polygons, use exterior angles ($\frac{360^\circ}{n}$). The arithmetic is cleaner.
3. Find isosceles triangles made by equal sides or radii, and copy their base angles.
4. With a circle, turn angles into arcs, add the arcs, and turn back.

Worked example

Regular pentagon $ABCDE$ has an equilateral triangle $ABF$ drawn inside it, with $F$ interior to the pentagon. What is the measure, in degrees, of $\angle EFC$?

(A) $150$ (B) $156$ (C) $162$ (D) $168$ (E) $174$

Solution. Step 1. Each angle of a regular pentagon is $\frac{3\cdot180^\circ}{5} = 108^\circ$.

Step 2. Triangle $ABF$ is equilateral, so $\angle FAB = 60^\circ$, leaving $\angle EAF = 108^\circ - 60^\circ = 48^\circ$.

Step 3. $AE = AB = AF$, so triangle $AEF$ is isosceles with $48^\circ$ at $A$. Its base angles share $180^\circ - 48^\circ = 132^\circ$, so $\angle AFE = 66^\circ$.

Step 4. The figure is symmetric across line $FD$, which swaps $A$ with $B$ and $E$ with $C$. So $\angle BFC = 66^\circ$ too.

Step 5. The angles $EFA$, $AFB$, $BFC$, $CFE$ go once around $F$, so they add up to $360^\circ$:
$$\angle EFC = 360^\circ - 66^\circ - 60^\circ - 66^\circ = 168^\circ.$$
The answer is $\boxed{\textbf{(D)}\ 168}$.

Pitfalls

• Using $(n-2)\cdot 180^\circ$ for the exterior angles. Exterior angles always add up to $360^\circ$.
• Deciding a triangle is isosceles, or a quadrilateral is a rectangle, because the drawing looks that way.
• Giving the arc when the question asks for the inscribed angle. You are off by a factor of two.

Traps that recur

• Halving the original apothem instead of the circumradius; the crease sits at half the distance from the center to a vertex, not to a side. (2023 AMC 10B #25)
• Trying to compute the angle by brute-force trigonometry or coordinates, which is slow and error-prone without a calculator. (2022 AMC 10B #20)
• Trying to solve for the six individual side lengths, which the data does not determine, instead of grouping them into the two triangle sides. (2021 AMC 10A #21)
• Assuming G is the midpoint of AC, or misreading which points F, G, H, J are, e.g. treating JH as a side of the inner pentagon rather than a diagonal. (2015 AMC 10B #22)