AQA · GCSE · Mathematics
Course version: 8300 · Version from Sept 2015
Apply the properties of angles at a point, angles at a point on a straight…
Full specification wording
Properties and constructions: apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles understand and use alternate and corresponding angles on parallel lines derive and use the sum of angles in a triangle (eg to deduce and use the angle sum in any polygon, and to derive properties of regular polygons)
Angle facts are applied in order.
Checked against the AQA 8300 specification (opens in a new tab) on .
- Point 3 of 15
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
Angle facts are applied in order. Angles at a point sum to 360°, angles at a point on a straight line sum to 180°, and vertically opposite angles are equal. On parallel lines, alternate angles are equal and corresponding angles are equal, which supports reasoning about figures containing a transversal. The sum of angles in a triangle is 180°; this is derived by drawing a line parallel to one side through the opposite vertex and using alternate angles, so the three angles of the triangle fit onto a straight line. From the triangle result, any polygon with n sides can be split into n − 2 triangles, giving an interior angle sum of (n − 2) × 180°. For a regular polygon, dividing this sum by n gives each interior angle, and exterior angles sum to 360°, so each exterior angle of a regular polygon is 360° ÷ n.
What earns the marks
- Applies angles at a point summing to 360° and angles at a point on a straight line summing to 180°.
- Uses the equality of vertically opposite angles when two straight lines cross.
- Identifies and uses alternate and corresponding angles on parallel lines, giving a reason for each equality.
- Derives and uses the sum of angles in a triangle as 180°, for example by drawing a parallel line to justify the result.
- Deduces the interior angle sum of any polygon with n sides as (n − 2) × 180° by splitting it into n − 2 triangles.
- Derives properties of regular polygons, including each interior angle as (n − 2) × 180° ÷ n and each exterior angle as 360° ÷ n.
Common mistakes and how to put them right
Confusing alternate and corresponding angles.
Put it right: Alternate angles lie on opposite sides of the transversal in a Z shape, while corresponding angles lie in matching positions in an F shape, and both are equal only when the lines are parallel.
Using 360° for the sum of angles in a triangle.
Put it right: The triangle sum is 180°, and 360° applies to angles at a point or to the exterior angles of a polygon.
Calculating a polygon interior angle sum as n × 180°.
Put it right: The correct sum is (n − 2) × 180°, because the polygon splits into n − 2 triangles.
Assuming vertically opposite angles are supplementary.
Put it right: Vertically opposite angles are equal, while adjacent angles on a straight line are the pair that sum to 180°.
Exam tips
- Give a reason for every angle fact used, naming the property such as alternate angles or angles on a straight line, so the reasoning is clear.
- Mark equal angles on the diagram as you find them, which reduces the chance of using the wrong pair of angles later.
- For polygon questions, identify n first, then substitute carefully into (n − 2) × 180° before dividing for a regular polygon.
- Check an answer against the diagram, for example confirming that an interior angle of a regular polygon is less than 180°.
Try the lesson
The start of the Properties and constructions: apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles understand and use alternate and corresponding angles on parallel lines derive and use the sum of angles in a triangle (eg to deduce and use the angle sum in any polygon, and to derive properties of regular polygons) lesson, made from checked teaching notes for this specification point.
Have a go first
Three angles meet at a point to form a complete turn. Two of the angles measure 124° and 89°. What is the size of the third angle?
Listen
Angle facts are applied in order. Angles at a point sum to 360°, angles at a point on a straight line sum to 180°, and vertically opposite angles are equal. On parallel lines, alternate angles are equal and corresponding angles are equal, which supports reasoning about figures containing a transversal. The sum of angles in a triangle is 180°; this is derived by drawing a line parallel to one side through the opposite vertex and using alternate angles, so the three angles of the triangle fit onto a straight line. From the triangle result, any polygon with n sides can be split into n − 2 triangles, giving an interior angle sum of (n − 2) × 180°. For a regular polygon, dividing this sum by n gives each interior angle, and exterior angles sum to 360°, so each exterior angle of a regular polygon is 360° ÷ n. A strong answer earns marks for these points. Applies angles at a point summing to 360° and angles at a point on a straight line summing to 180°. Uses the equality of vertically opposite angles when two straight lines cross. Identifies and uses alternate and corresponding angles on parallel lines, giving a reason for each equality. Derives and uses the sum of angles in a triangle as 180°, for example by drawing a parallel line to justify the result. Exam tip: Give a reason for every angle fact used, naming the property such as alternate angles or angles on a straight line, so the reasoning is clear.
Key words
Say what each word means before you turn the card.
Angles at a point and on a straight line
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Vertically opposite angles
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Alternate and corresponding angles on parallel lines
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Angle sums in triangles, polygons and regular polygons
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See every definition
- Angles at a point and on a straight line
- Angles at a point sum to 360°, while angles at a point on a straight line sum to 180°.
- Vertically opposite angles
- When two straight lines cross, these angles are equal.
- Alternate and corresponding angles on parallel lines
- On parallel lines, alternate angles are equal and corresponding angles are equal, which supports reasoning about figures containing a transversal.
- Angle sums in triangles, polygons and regular polygons
- The sum of angles in a triangle is 180°, any polygon with n sides has an interior angle sum of (n − 2) × 180°, and exterior angles sum to 360°.
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Test your understanding of Properties and constructions: apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles understand and use alternate and corresponding angles on parallel lines derive and use the sum of angles in a triangle (eg to deduce and use the angle sum in any polygon, and to derive properties of regular polygons)
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