Angle facts

• Angles on a straight line add up to $180^\circ$; angles at a point add up to $360^\circ$.
• Vertically opposite angles are equal.
• Parallel lines: corresponding angles are equal, alternate angles are equal, co-interior angles add up to $180^\circ$.
• Angles of a triangle add up to $180^\circ$; exterior angle $=$ sum of the two opposite interior angles.

Polygons ($n$ sides)

$$\text{Sum of interior angles} = (n - 2)\times 180^\circ \qquad \text{Sum of exterior angles} = 360^\circ$$ $$\text{Each interior angle (regular)} = \frac{(n - 2)\times 180^\circ}{n} \qquad \text{Each exterior angle (regular)} = \frac{360^\circ}{n}$$ $$\text{Number of diagonals} = \frac{n(n - 3)}{2}$$

Circle theorems

1. The angle at the centre is twice the angle at the circumference on the same arc.
2. The angle in a semicircle is $90^\circ$.
3. Angles in the same segment are equal.
4. Opposite angles of a cyclic quadrilateral add up to $180^\circ$; an exterior angle equals the interior opposite angle.
5. A tangent is perpendicular to the radius at the point of contact.
6. Tangents drawn from an external point are equal in length.
7. Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
8. Intersecting chords: $AP \times PB = CP \times PD$. Tangent and secant: $PT^2 = PA \times PB$.