BLOG

The perpendicular bisector of a line cuts it in half at a right angle, and a pair of compasses finds it exactly. Diagnose six attempted constructions in an interactive playground.

Bisecting an angle splits it into two equal halves using compasses only, with no protractor and no measuring. Work out the half-angle for four constructions in an interactive playground.

Given three side lengths, a ruler and compasses build the triangle exactly: draw one side, then swing an arc from each end. Find the apex for four triangles in an interactive playground.

Complementary and supplementary are not shapes or positions. They are a claim about a total: two angles that add to 90° are complementary, two that add to 180° are supplementary. Learn to read which total a figure is asking for, then find the missing angle yourself in an interactive playground.

A linear pair is the first angle name where position matters as much as size. The two angles must sit next to each other, and their outer arms must form a straight line — which is what forces them to add to 180°. Sort the real linear pairs from the convincing near-misses in an interactive playground.

When two straight lines cross they make four angles, and the pairs facing each other across the crossing are always equal. That is not a coincidence to memorise — it follows from linear pairs in two steps. Prove it to yourself, then use it to fill in a crossing from one given angle in an interactive playground.

A transversal is a line that cuts across two others. When the two lines it cuts are parallel, something remarkable happens: the two crossings become identical copies, so all eight angles are one of only two sizes. See exactly what parallelism buys you — and what happens without it — in an interactive playground.

Corresponding angles sit in matching positions at the two crossings; alternate angles sit on opposite sides of the transversal. Both are equal when the lines are parallel, and both are just ways of asking which of the two available sizes an angle is. Name them on a real diagram in an interactive playground.

Every other named pair on parallel lines turns out to be equal. Co-interior angles are the exception: they add to 180° instead. The reason is short, and once you see it the exception stops feeling arbitrary. Find the missing partner on four diagrams in an interactive playground.

Extend any side of a triangle and the angle that opens up outside it is exactly the sum of the two angles at the far corners. It is the fastest shortcut in triangle geometry and it falls straight out of two rules you already have. Use it on four triangles in an interactive playground.

Not every three lengths will close into a triangle. The test is one addition: the two shorter sides together must be longer than the longest one, and longer is doing real work — equal is not enough. See exactly where the failure happens in an interactive playground.

In an isosceles triangle the two angles opposite the equal sides are equal too. That is not a second fact bolted on to the first — the equal sides force it, and folding the triangle in half shows why. One given angle is then enough to find the rest, which you can practise in an interactive playground.

Congruent means identical in shape and size. You do not need all six measurements to be sure of it — three of the right kind will do. Three sides work, and so do two sides with the angle between them, while two very similar-looking sets of facts do not. Sort them in an interactive playground.

Angles alone can never prove two triangles identical, because they say nothing about size. Add a single side and they can: ASA and AAS both work, and right-angled triangles get a rule of their own in RHS. Sort the rules from the near-misses in an interactive playground.

Pythagoras' theorem is usually met as a formula, which hides what it means. Draw a square on each side of a right-angled triangle and the big one holds exactly as much as the two small ones together. Work with the squares themselves in an interactive playground before the formula appears.

When both short sides of a right-angled triangle are known, the longest side comes from three steps: square each, add, take the square root. The adding is what marks this case out from the next one. Work through six triangles in an interactive playground.

When the longest side is known and a shorter one is not, Pythagoras' theorem runs backwards: subtract instead of adding. The trap is the order — the big square always goes first. Practise the direction most often got wrong in an interactive playground.

Pythagoras' theorem works in reverse. If the two shorter sides squared and added come to exactly the longest side squared, the triangle must contain a right angle — no protractor needed. And when they do not balance, the direction of the miss tells you what kind of triangle it is. Test six in an interactive playground.

Every four-sided shape has angles totalling 360°, however lopsided it looks. The reason is a single diagonal: it splits any quadrilateral into two triangles, and two lots of 180° make 360°. Find the missing angle on six of them in an interactive playground.

A triangle holds 180° and a quadrilateral 360°, and the pattern does not stop there. Any polygon can be cut into triangles from a single corner, and the number of triangles is always two fewer than the number of sides. Count them and multiply, in an interactive playground.