Topics / Geometry

Angles & Polygons

Angle chasing, polygon angle sums, regular polygons, parallel lines

Angles & PolygonsAMC 10 Step by Step

Name: Date: September 13, 2026

Geometry · 18 problems have this as their main idea

primary-topic problems (1.4% of all)

What you need to know

Interior angles of an $n$ -gon sum to $(n-2)\cdot 180^\circ$ ; exterior angles of any convex polygon sum to $360^\circ$ . A regular $n$ -gon has exterior angle $360^\circ/n$ and interior angle $180^\circ - 360^\circ/n$ .
Parallel lines with a transversal give equal alternate interior angles and supplementary co-interior angles. An exterior angle of a triangle equals the sum of the two remote interior angles.
Base angles of an isosceles triangle are equal; the equal sides of a regular polygon hide many isosceles triangles.
An inscribed angle is half its intercepted arc; a central angle equals its arc.
A convex $n$ -gon has $\frac{n(n-3)}{2}$ diagonals.

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 for $n$ in $360/n$ .
Problems 10–16: a regular pentagon, hexagon, or octagon with diagonals or an attached equilateral triangle or square; the trick is spotting isosceles triangles.

Standard approaches

Label unknown angles with variables and write triangle and polygon sum equations.
Use exterior angles ( $360^\circ/n$ ) for regular polygons; the arithmetic is cleaner.
Find isosceles triangles formed by equal sides or radii and copy their base angles.
With a circle present, convert angles to arcs, add arcs, and convert 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 $\angle EFD$ ?

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

Solution. Each interior angle of the pentagon is $108^\circ$ . Since $\angle FAB = 60^\circ$ , $\angle EAF = 108^\circ - 60^\circ = 48^\circ$ . Now $AE = AB = AF$ , so triangle $AEF$ is isosceles with apex $48^\circ$ and base angles $\angle AFE = 66^\circ$ . By symmetry $\angle BFD = 66^\circ$ . The angles around $F$ sum to $360^\circ$ , so
$\angle EFD = 360^\circ - 66^\circ - 60^\circ - 66^\circ = 168^\circ.$
The answer is $\boxed{\textbf{(D)}\ 168}$ .

Pitfalls

Using $(n-2)\cdot 180$ for the exterior angle sum.
Assuming a triangle is isosceles or a quadrilateral is a rectangle because the sketch looks that way.
Reporting an arc instead of the inscribed angle (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)

Problems, easiest first

2012 AMC 10A · #4Angles & Polygons

Let $\angle ABC = 24^\circ$ and $\angle ABD = 20^\circ$ . What is the smallest possible degree measure for angle $CBD$ ?

2021 AMC Fall 10A · #7Angles & Polygons

As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$ . Point $F$ lies on $\overline{AD}$ so that $DE=DF$ , and $ABCD$ is a square. What is the degree measure of $\angle AFE$ ?

2011 AMC 10B · #7Angles & Polygons

The sum of two angles of a triangle is $6/5$ of a right angle, and one of these two angles is $30^{\circ}$ larger than the other. What is the degree measure of the largest angle in the triangle?

2009 AMC 10B · #9Angles & Polygons

Segment $BD$ and $AE$ intersect at $C$ , as shown, $AB=BC=CD=CE$ , and $\angle A = \frac 52 \angle B$ . What is the degree measure of $\angle D$ ?

2007 AMC 10B · #7Angles & Polygons

All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A= \angle B = 90^\circ.$ What is the degree measure of $\angle E?$

2007 AMC 10B · #15Angles & Polygons

The angles of quadrilateral $ABCD$ satisfy $\angle A=2 \angle B=3 \angle C=4 \angle D.$ What is the degree measure of $\angle A,$ rounded to the nearest whole number?

2007 AMC 10A · #8Angles & Polygons

Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$ . Point $D$ is inside triangle $ABC$ , angle $ABC$ measures $40$ degrees, and angle $ADC$ measures $140$ degrees. What is the degree measure of angle $BAD$ ?

2025 AMC 10A · #6Angles & Polygons

In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle $20^{\circ}$ -angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?

2019 AMC 10A · #14Angles & Polygons

For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$ ?

2010 AMC 10A · #14Angles & Polygons

Triangle $ABC$ has $AB=2 \cdot AC$ . Let $D$ and $E$ be on $\overline{AB}$ and $\overline{BC}$ , respectively, such that $\angle BAE = \angle ACD$ . Let $F$ be the intersection of segments $AE$ and $CD$ , and suppose that $\triangle CFE$ is equilateral. What is $\angle ACB$ ?

2006 AMC 10A · #19Angles & Polygons

How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?

2001 AMC 10 · #20Angles & Polygons

A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length $2000$ . What is the length of each side of the octagon?

2022 AMC 10B · #20Angles & Polygons

Let $ABCD$ be a rhombus with $\angle{ADC} = 46^{\circ}$ . Let $E$ be the midpoint of $\overline{CD}$ , and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$ . What is the degree measure of $\angle{BFC}$ ?

2021 AMC 10A · #21Angles & Polygons

Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$ , and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$ . The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\sqrt{p}$ , where $m, n,$ and $p$ are positive integers and $p$ …

2015 AMC 10B · #22Angles & Polygons

In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$ . What is $FG+JH+CD$ ?

2009 AMC 10B · #24Angles & Polygons

The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior …

2008 AMC 10B · #24Angles & Polygons

Quadrilateral $ABCD$ has $AB = BC = CD$ , angle $ABC = 70^\circ$ and angle $BCD = 170^\circ$ . What is the measure of angle $BAD$ ?

2023 AMC 10B · #25Angles & Polygons

A regular pentagon with area $1+\sqrt5$ is printed on paper and cut out. All five vertices are folded to the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?