Geometry and circle theorems
Angle facts, parallel lines, polygons, circle theorems, similarity, symmetry and constructions.
📌 Key points
- Angles on a straight line add to 180° and angles around a point add to 360°. Vertically opposite angles are equal.
- With parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior angles add to 180°.
- Angles in a triangle add to 180° and in a quadrilateral to 360°. An exterior angle of a triangle equals the sum of the two opposite interior angles.
- Polygons: interior angles add to (n − 2) × 180°. Exterior angles of any polygon add to 360°, so each exterior angle of a regular n-sided polygon is 360° ÷ n.
- The angle in a semicircle is 90°. A tangent is perpendicular to the radius at the point of contact. Two tangents from an outside point are equal in length.
- The angle at the centre is twice the angle at the circumference standing on the same arc. Angles in the same segment are equal.
- Opposite angles of a cyclic quadrilateral add to 180°. Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
- Similar shapes: if lengths are multiplied by k, areas are multiplied by k2 and volumes by k3. Congruent shapes are identical in shape and size.
- A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.
- Constructions: the perpendicular bisector of AB gives points the same distance from A and B; an angle bisector gives points the same distance from two lines.
📖 Key terms
- Alternate angles
- Equal angles on opposite sides of a line crossing two parallel lines.
- Corresponding angles
- Equal angles in matching positions where a line crosses two parallel lines.
- Co-interior angles
- Angles between two parallel lines on the same side of the crossing line; they add to 180°.
- Cyclic quadrilateral
- A quadrilateral with all four vertices on a circle.
- Tangent
- A straight line that touches a circle at exactly one point.
- Chord
- A straight line joining two points on a circle.
- Similar
- Same shape, with all lengths in the same ratio; angles are equal.
- Congruent
- Exactly the same shape and size.
✏️ Worked example
Each interior angle of a regular polygon is 156°. How many sides does it have?
- Interior and exterior angles add to 180°, so each exterior angle is 180° − 156° = 24°.
- Exterior angles of a polygon add to 360°.
- Number of sides = 360 ÷ 24 = 15.
- Check: (15 − 2) × 180° = 2340° and 2340 ÷ 15 = 156°. Answer: 15 sides.
Give a geometric reason for each angle using the proper words, such as “alternate angles” or “angle in a semicircle”. “Z angles” or “F angles” do not earn the reason mark.
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