Geometry and trigonometry

• Angles on a straight line sum to 180 degrees; around a point, to 360.
• Vertically opposite angles are equal.
• Parallel lines give equal corresponding angles, equal alternate angles, and co-interior angles summing to 180.
• A triangle's angles sum to 180; an exterior angle equals the sum of the two opposite interior angles.
• A polygon with n sides has interior angles summing to 180 times n minus 2, and exterior angles always summing to 360.

Congruent shapes are identical in size and shape. The standard tests are side-side-side, side-angle-side, angle-side-angle and right angle with hypotenuse and one side. Notably side-side-angle is not a valid test, since two different triangles can satisfy it.

Similar shapes have the same angles and proportional sides. If lengths scale by a factor k, areas scale by k squared and volumes by k cubed. That relationship answers a great many problems quickly and is a frequent source of errors when people scale area by k directly.

In a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides. The converse also holds, which makes it a test for whether an angle is right.

The trigonometric ratios extend this to angles. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. These are ratios, so they depend only on the angle, which is exactly why they work for every similar triangle at once.

For non-right triangles, the sine rule relates each side to the sine of its opposite angle and is used when you have a matched side and angle pair. The cosine rule generalises Pythagoras and is used when you have three sides, or two sides with the angle between them.

The sine rule has an ambiguous case: when finding an angle, two angles between 0 and 180 can share the same sine. Check whether both produce a valid triangle rather than accepting the calculator's answer automatically.

Circle theorems repay memorisation because they turn hard-looking diagrams into one-line deductions. The angle at the centre is twice the angle at the circumference on the same arc. Angles in the same segment are equal. The angle in a semicircle is a right angle. Opposite angles of a cyclic quadrilateral sum to 180. A tangent meets a radius at a right angle.

When a circle problem stalls, add the radii to the diagram. Radii are equal, so they create isosceles triangles, and the base angles of those triangles are usually what the question wants.