Geometry & measures

Angles, Polygons and Bearings

Year 10 · Year 11

✓Apply angle rules for angles on a straight line, at a point, and vertically opposite angles.
✓Use angle properties of parallel lines, including alternate, corresponding, and interior angles.
✓Calculate the sum of interior and exterior angles of polygons, and find individual angles in regular polygons.
✓Understand and calculate bearings, including reverse bearings, ensuring correct notation.

Key concepts

Basic Angle Rules

These fundamental rules apply to all angles: - Angles on a straight line sum to 180°. - Angles at a point (or angles around a point) sum to 360°. - Vertically opposite angles are equal. These are formed when two straight lines intersect.

Angles in Parallel Lines

When a transversal line intersects two parallel lines, specific angle relationships are formed: - Corresponding angles are equal (they form an 'F' shape). - Alternate angles are equal (they form a 'Z' shape). - Interior angles (also known as consecutive or allied angles) sum to 180° (they form a 'C' or 'U' shape). These are the angles between the parallel lines and on the same side of the transversal.

Interior Angles of Polygons

A polygon is a closed 2D shape with straight sides. The sum of the interior angles of an n-sided polygon can be calculated using a formula. For a regular polygon, all interior angles are equal.

Sum of interior angles = (n - 2) × 180° Each interior angle of a regular n-sided polygon = ((n - 2) × 180°) / n

Exterior Angles of Polygons

An exterior angle of a polygon is formed by extending one side of the polygon. The sum of the exterior angles of any convex polygon is always 360°. For a regular polygon, all exterior angles are equal. Also, an interior angle and its adjacent exterior angle always sum to 180°.

Sum of exterior angles = 360° Each exterior angle of a regular n-sided polygon = 360° / n Interior angle + Exterior angle = 180°

Bearings

Bearings are used to describe the direction of one point relative to another. They are always measured: 1. From the North line. 2. Clockwise. 3. As a three-figure number (e.g., 045°, 120°, 300°). If the angle is less than 100°, a leading zero is used.

If the bearing of B from A is θ: - If θ < 180°, the bearing of A from B (reverse bearing) = θ + 180°. - If θ > 180°, the bearing of A from B (reverse bearing) = θ - 180°.

Key facts to remember

1Angles on a straight line sum to 180°.
2Angles at a point sum to 360°.
3Vertically opposite angles are equal.
4Alternate angles are equal (Z-shape).
5Corresponding angles are equal (F-shape).
6Interior angles between parallel lines sum to 180° (C-shape).
7The sum of interior angles of an n-sided polygon is (n - 2) × 180°.
8The sum of exterior angles of any polygon is 360°.
9Bearings are measured clockwise from North and are always written with three figures.

Worked examples

Example 1

In the diagram, lines AB and CD are parallel. Line EF is a transversal. Angle AEF = 65°. Find the size of angle EFG and angle CFE. Give reasons for your answers.

IAngle EFG and angle AEF are alternate angles. Since AB is parallel to CD, alternate angles are equal.

IITherefore, Angle EFG = Angle AEF = 65° (Alternate angles are equal).

IIIAngle CFE and angle AEF are interior angles. Since AB is parallel to CD, interior angles sum to 180°.

IVTherefore, Angle CFE + Angle AEF = 180°.

VAngle CFE + 65° = 180°.

VIAngle CFE = 180° - 65° = 115° (Interior angles sum to 180°).

Answer

Angle EFG = 65°, Angle CFE = 115°

Always state the geometric reasons for your angle calculations in an exam.

Example 2

A regular polygon has an exterior angle of 30°. Calculate: a) The number of sides of the polygon. b) The sum of its interior angles.

Ia) For any regular polygon, the sum of the exterior angles is 360°.

IISince all exterior angles of a regular polygon are equal, the number of sides (n) can be found by dividing 360° by the size of one exterior angle.

IIINumber of sides (n) = 360° / Exterior angle.

IVn = 360° / 30° = 12.

Vb) The sum of the interior angles of an n-sided polygon is given by the formula (n - 2) × 180°.

VISubstitute n = 12 into the formula:

VIISum of interior angles = (12 - 2) × 180°.

VIIISum of interior angles = 10 × 180° = 1800°.

Answer

a) Number of sides = 12 b) Sum of interior angles = 1800°

Alternatively for part b), you could find the interior angle first (180° - 30° = 150°) and then multiply by the number of sides (150° × 12 = 1800°).

Example 3

The bearing of point B from point A is 055°. Calculate the bearing of point A from point B.

IDraw a diagram showing points A and B with a North line at A.

IIMark the bearing of B from A as 055° (clockwise from North at A).

IIIDraw a North line at B. The line AB is a transversal between two parallel North lines.

IVThe angle between the North line at B and the line BA (measured anti-clockwise from North) is an alternate angle to the bearing of B from A.

VAlternatively, use the reverse bearing rule: If the bearing of B from A is θ, then the bearing of A from B is θ + 180° (if θ < 180°) or θ - 180° (if θ > 180°).

VIHere, θ = 055°, which is less than 180°.

VIIBearing of A from B = 055° + 180° = 235°.

Answer

The bearing of A from B is 235°.

Always draw a clear diagram with North lines to help visualise bearing problems, especially for reverse bearings.

Common mistakes

✗Confusing alternate and corresponding angles, or mixing up their properties.
✗Forgetting to state the geometric reasons for angle calculations in exam questions, which often leads to loss of marks.
✗Incorrectly applying the polygon angle formulas, especially using 'n' for exterior angles when it should be 360/n.
✗Not writing bearings with three figures (e.g., writing 70° instead of 070°).
✗Incorrectly calculating reverse bearings, for example, always adding 180° even when the initial bearing is greater than 180°.

Exam tips

★Always state the geometric reasons (e.g., 'alternate angles are equal', 'angles on a straight line') for every step in your angle calculations to gain full marks.
★Draw clear diagrams for bearing problems, including North lines at each relevant point, to help visualise the angles and directions.
★For polygon questions, check if the polygon is regular. If it is, all interior angles are equal and all exterior angles are equal.
★When dealing with regular polygons, using the exterior angle property (sum = 360°) can often simplify calculations, as it's easier to find the number of sides or the exterior angle first.

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