Geometry & measures
Angles, Parallel Lines, Polygons and Bearings
Year 7 · Year 8 · Year 9
- ✓Identify and use angle properties associated with parallel lines, including corresponding, alternate, and interior angles.
- ✓Calculate the sum of interior angles and the size of each interior/exterior angle of regular and irregular polygons.
- ✓Solve problems involving angles in polygons, including quadrilaterals and triangles.
- ✓Understand and use three-figure bearings to describe directions.
- ✓Apply angle rules and properties to solve geometric problems in various contexts.
Key concepts
These are fundamental rules for angles that apply in various geometric situations.
Angles that lie on a straight line sum to 180°.
Angles that meet at a point sum to 360°.
When two straight lines intersect, the angles opposite each other at the intersection point are called vertically opposite angles. They are equal.
When two parallel lines are intersected by a transversal line, specific pairs of angles are formed with special properties.
These are angles in the same relative position at each intersection. They form an 'F' shape. Corresponding angles are equal.
These are angles on opposite sides of the transversal and between the parallel lines. They form a 'Z' shape. Alternate angles are equal.
These are angles on the same side of the transversal and between the parallel lines. They form a 'C' or 'U' shape. Interior angles sum to 180°.
Polygons are 2D shapes with straight sides. The sum of their interior and exterior angles follows specific rules.
The total sum of all interior angles in any polygon with 'n' sides.
For a regular polygon (all sides and angles equal), this is the size of one interior angle.
The total sum of all exterior angles (formed by extending one side) of any polygon.
For a regular polygon, this is the size of one exterior angle.
An interior angle and its adjacent exterior angle always form a straight line.
Bearings are used to describe the direction of one point from another. They are measured clockwise from the North line and are always written as three figures.
Key facts to remember
- 1Corresponding angles are equal (often forming an 'F' shape).
- 2Alternate angles are equal (often forming a 'Z' shape).
- 3Interior angles (also known as allied or conjoined angles) sum to 180° (often forming a 'C' or 'U' shape).
- 4Angles on a straight line sum to 180°.
- 5Angles around a point sum to 360°.
- 6Vertically opposite angles are equal.
- 7The sum of interior angles of an n-sided polygon is (n - 2) × 180°.
- 8The sum of exterior angles of any polygon is always 360°.
- 9An interior angle and its adjacent exterior angle sum to 180°.
- 10Bearings are measured clockwise from North and are always given as three figures (e.g., 045°, 180°, 315°).
Worked examples
Example 1
In the diagram, lines AB and CD are parallel. A transversal line intersects AB at E and CD at F. Angle BEF = 110°. Find the size of angle CFE (x) and angle DFG (y), where G is a point on the transversal above F. Give reasons for your answers.
Answer
x = 70°, y = 70°
There can often be multiple correct ways to find angles; choose the most direct path and always state your reasons.
Example 2
A regular octagon has an interior angle of 135°. Calculate the sum of its interior angles and the size of each exterior angle.
Answer
Sum of interior angles = 1080°, Each exterior angle = 45°
For regular polygons, all interior angles are equal and all exterior angles are equal.
Example 3
A boat travels from port P to port Q on a bearing of 060°. What is the bearing of port P from port Q?
Answer
240°
Always draw a diagram with North lines at both points when working with 'back bearings'.
Common mistakes
- ✗Confusing corresponding, alternate, and interior angle rules, especially when diagrams are rotated or complex.
- ✗Incorrectly using the polygon angle formulas, particularly mixing up interior and exterior angle calculations or using the wrong 'n' value.
- ✗Forgetting to write bearings as three figures (e.g., writing 45° instead of 045°).
- ✗Measuring bearings anti-clockwise or from a line other than the North line.
- ✗Assuming lines are parallel or shapes are regular when not explicitly stated or indicated by notation (e.g., arrows for parallel lines, tick marks for equal sides).
Exam tips
- ★Always state the geometric reason for each step when calculating angles (e.g., 'alternate angles are equal', 'angles on a straight line'). This is crucial for gaining full marks.
- ★Draw clear diagrams, especially for bearings problems, and mark all known angles and North lines. This helps visualise the problem.
- ★For polygon problems, remember that the sum of exterior angles is always 360°, which can be a quick way to find individual exterior angles for regular polygons or to check your interior angle calculations.
- ★When finding a 'back bearing' (e.g., bearing of A from B when you know B from A), draw parallel North lines at both points and use parallel line angle rules to help determine the angle.
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