Geometry
Geometry
Geometry tests angle reasoning in given figures, the properties of polygons and circles, congruence and similarity, and three-figure bearings, together with accurate ruler-and-compass construction. Marks are awarded for reasons, so every step should name the angle fact you used.
Angles and polygons
- Angles on a straight line sum to 180°; angles at a point sum to 360°; vertically opposite angles are equal.
- With parallel lines, alternate and corresponding angles are equal and co-interior angles sum to 180°.
- Interior angle sum of an n-sided polygon = (n − 2) × 180°. Exterior angles sum to 360°, so each exterior angle of a regular n-gon is 360°/n.
Circle theorems
- Angle at the centre = 2 × angle at the circumference standing on the same arc.
- Angle in a semicircle = 90°; angles in the same segment are equal.
- Opposite angles of a cyclic quadrilateral sum to 180°.
- A tangent is perpendicular to the radius at the point of contact; the alternate segment theorem applies.
Congruence and similarity
- Congruence tests: SSS, SAS, ASA, RHS.
- Similar figures have corresponding lengths in the same ratio; areas are in the square of that ratio.
Bearings
- Three figures, measured clockwise from north, so 45° is written 045°.
Worked example. Each interior angle of a regular polygon is 150°. Find the number of sides. Exterior angle = 180° − 150° = 30° (interior and exterior angles lie on a straight line). n = 360° ÷ 30° = 12. The polygon has 12 sides.
Traps that cost marks.
- Giving the numerical answer without the reason.
- Confusing interior and exterior angles, or using 360/n as the interior angle.
- Measuring a bearing anticlockwise or from the wrong north line.
- Assuming figures are drawn to scale when the question does not say so.