Angle Properties and Rules
This topic covers the fundamental rules of angles and shapes, which are essential for solving geometric problems without a calculator. You will use these properties to find unknown angles in diagrams involving lines, triangles, and polygons.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Angles at a point sum to 360°, and angles on a straight line sum to 180°.
- When a transversal crosses parallel lines: alternate angles (Z-shape) are equal, corresponding angles (F-shape) are equal, and co-interior angles (C-shape) sum to 180°.
- Vertically opposite angles, formed by the intersection of two straight lines, are always equal.
- The sum of interior angles in a triangle is 180°. For a quadrilateral, it's 360°.
- The sum of the exterior angles of any convex polygon is always 360°, regardless of the number of sides.
- The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Formulae
Sum of interior angles = 180 × (n - 2) To find the total sum of all interior angles in any n-sided polygon.
Single interior angle of a regular polygon = 180 - (360 / n) A quick way to find the size of one interior angle in a regular n-sided polygon. This is often faster than calculating the total sum first.
Definitions
- Transversal
- A straight line that intersects two or more other lines.
- Regular Polygon
- A polygon where all side lengths are equal and all interior angles are equal.
- Exterior Angle
- The angle formed between a side of a polygon and the extension of its adjacent side.
Worked example
A regular hexagon and a square share a common side, as shown in the diagram which is not to scale. Find the angle marked x.
- 1
First, find the size of one interior angle of a regular hexagon (n=6).
Using the formula:
180 - (360/n) = 180 - (360/6) = 180 - 60 = 120°.
- 2
Next, identify the interior angle of the square.
All angles in a square are 90°.
- 3
The angle x, the hexagon's angle, and the square's angle meet at a single point.
The sum of angles around a point is 360°.
- 4
Set up the equation:
x + 120 + 90 = 360.
- 5
Solve for x:
x = 360 - 210 = 150°
Answer: 150°
Common mistakes
- ×Confusing the different types of angles associated with parallel lines (alternate, corresponding, co-interior). Sketching Z, F, and C shapes can help distinguish them.
- ×Incorrectly applying formulae for regular polygons to irregular polygons. The 'single interior angle' formula only works if all angles are equal.
- ×Forgetting that the sum of exterior angles is a constant 360° for any polygon, and mistakenly trying to use the (n-2) formula.
- ×Mixing up supplementary (add to 180°) and complementary (add to 90°) angles.
No-calculator tips
- ✓Memorise the interior angle sums for common shapes: Triangle (180), Quadrilateral (360), Pentagon (540), Hexagon (720). Each step up adds 180°.
- ✓To find an interior angle of a regular polygon, it's often faster to calculate the exterior angle (360/n) first and then subtract it from 180.
- ✓In complex diagrams, extend lines to create more intersections. This can reveal simpler angle relationships like alternate or corresponding angles that were not immediately obvious.