Lines, triangles, polygons, circles, and coordinate geometry.

Geometry deals with the properties of shapes, angles, and positions. Most questions can be solved by recalling a small set of facts and formulas and applying them carefully to the figure.

Lines and angles

• Angles on a straight line add up to 180 degrees, and angles around a point add up to 360 degrees.
• Vertically opposite angles are equal.
• When a transversal cuts two parallel lines, corresponding angles and alternate angles are equal, and co-interior angles add up to 180 degrees.

Triangles

• The angles of a triangle add up to 180 degrees.
• Pythagoras theorem: in a right triangle, (hypotenuse)^2 = (base)^2 + (height)^2.
• Area = 1/2 x base x height.
• For similar triangles, the sides are in the same ratio, and the areas are in the ratio of the squares of the sides.

Polygons

• The sum of the interior angles of an n-sided polygon is (n - 2) x 180 degrees.
• The sum of the exterior angles of any polygon is 360 degrees.
• In a regular polygon, each exterior angle is 360 / n degrees and each interior angle is 180 degrees minus the exterior angle.

Circles

• Circumference = 2 x pi x r, and area = pi x r^2.
• A perpendicular from the centre to a chord bisects the chord.
• The angle at the centre is twice the angle at the circumference on the same arc.
• The angle in a semicircle is 90 degrees.

Coordinate geometry

• Distance between (x1, y1) and (x2, y2) = √((x2 - x1)^2 + (y2 - y1)^2).
• Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2).
• Slope of a line = (y2 - y1) / (x2 - x1).
• Equation of a line through (x1, y1) with slope m: y - y1 = m(x - x1).

How to solve questions

1. Draw or visualise the figure and mark the given values.
2. Identify the shape and the fact or formula that applies.
3. Substitute the values and solve step by step.
4. Check that the answer is sensible, for example that angles and lengths are positive.