Triangles
A triangle is the simplest polygon. Why do so many of geometry’s theorems turn out to be about it?
Where you are 18 inside
What this area is about
Three lengths fix a triangle, and every polygon can be cut into triangles. So most of plane geometry comes back to them: their angles, their sides, when two match, and the right triangle.
How these fit together
Parts. Parts of a triangle names vertices, sides and angles. Special segments in a triangle (medians, altitudes, angle bisectors, perpendicular bisectors) meet at the triangle centers.
Angles. The triangle angle sum is 180^\circ; the exterior angle theorem follows, in the same star.
Sides. The triangle inequality: any two sides together are longer than the third. Sides and angles in order puts the longest side opposite the largest angle, and the hinge theorem compares two triangles.
Congruence. Congruent triangles and SSS, SAS, ASA, AAS, three postulates and a theorem, prove when two triangles match. Isosceles and equilateral triangles turn equal sides into equal angles, and back.
Right triangles. The Pythagorean theorem, a^2 + b^2 = c^2 for legs a, b and hypotenuse c, with its applications. The converse of the Pythagorean theorem tests for a right angle; Pythagorean triples such as 3, 4, 5 fit it in whole numbers. Special right triangles, 45-45-90 and 30-60-90, have side ratios you can read off. Geometric means in a right triangle start from the altitude to the hypotenuse, which splits it into two similar triangles.
A path through it
Parts, angle sum, triangle inequality; then the Pythagorean theorem, its converse and triples. Then congruent triangles with the criteria, isosceles triangles, sides and angles in order, the hinge theorem and special right triangles. Special segments and centers last; geometric means wait for similarity.
This is the ground for geometric proof and similarity.