Angle sum of a triangle and polygons

Draw any triangle, as long and thin as you like, and its three angles still add to 180°. A heptagon's angles add to 900°.

Why a triangle makes 180°

OpenStax calls it one of the most important properties of triangles: the interior angles add up to 180°. You can test it with paper. Cut out a triangle, tear off its three corners and lay them side by side with their points together. They fill a straight line exactly. The lab below does the same with the Line them up button, for any triangle you drag into shape.

The proof uses the previous unit. Through the top corner C, draw a line parallel to the bottom side AB. The two sides from C act as transversals. The angle between the new line and side CA is an alternate interior angle to angle A, so it equals A; on the other side the same happens with B. Now the three angles at C, namely A, then C itself, then B, sit along the new straight line. A straight angle is 180°, so A + B + C = 180°.

What follows from 180°

A triangle can have at most one right angle or one obtuse angle, because two of them would already reach 180° or more. In a right triangle the two acute angles add to 90°, so they are complementary. In an equilateral triangle all three angles are equal, so each is 180 ÷ 3 = 60°. In an isosceles triangle, the two angles at the base are equal; OpenStax credits Thales with this discovery as well. If the top angle of an isosceles triangle is 40°, each base angle is (180 − 40) ÷ 2 = 70°.

An exterior angle is made by extending one side past a corner. It forms a linear pair with the interior angle there, so it equals 180° minus that angle, and since the three interior angles add to 180°, it also equals the sum of the two far angles. With A = 50° and B = 60°, the exterior angle at C is 110°, without ever finding C (which is 70°).

Polygons

Any polygon can be cut into triangles by drawing diagonals from one corner. A polygon with n sides splits into n − 2 triangles, and their angles together make up the angles of the polygon. That gives the formula OpenStax states: the interior angles of an n-sided polygon add to S = (n − 2) × 180°. A quadrilateral gives 360°, a pentagon 540°, a hexagon 720°, a heptagon 900°. In a regular polygon all the angles are equal, so each one is S ÷ n: 108° for a regular pentagon, 120° for a regular hexagon like a honeycomb cell.

There is a second way to see polygon sums. Walk all the way round the edge of a polygon. At each corner you turn through its exterior angle, and by the time you are back where you started you have made exactly one full turn. So the exterior angles of any convex polygon add to 360°, whatever the number of sides. For a regular hexagon each exterior angle is 360 ÷ 6 = 60°, and each interior angle is 180 − 60 = 120°, which agrees with the formula.