Triangles and quadrilaterals: constructing and characterising

Until now, you could recognise a triangle or a square at a glance. In this lesson, we level up: constructing them precisely, and characterising them, meaning identifying them by their exact properties rather than their looks. A geometer never says "it looks like a rhombus", they check: 4 sides of equal length? Then it is a rhombus, proved!

Let us start with triangles. The equilateral triangle has 3 sides of the same length. The isosceles triangle has at least 2 equal sides: it has a nice symmetry, like a well-balanced pointy hat. The right triangle has a right angle, the one you check with a set square. To construct a triangle when you know its 3 lengths, the star tool is the compass: it transfers lengths without ever making a mistake.

On the quadrilateral side, here is the whole family. The parallelogram has its opposite sides parallel in pairs. The rectangle adds 4 right angles. The rhombus bets everything on the sides: 4 sides of equal length. And the square? It is the champion: it has it all, 4 right angles and 4 equal sides. So a square is both a special rectangle and a special rhombus.

Keep this idea of nested families in mind: the more properties a shape has, the more special it is. In the exercises, you will have to pick the most precise characterisation, and avoid the classic trap of thinking a square is not a rectangle. Grab your ruler, compass and set square: let us build!