Foundation - Angles - GCSE Mathematics

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Foundation - Angles

infoWhy this? Angle properties let us deduce measurements from geometric relationships and justify conclusions rather than rely on appearance. Working with parallel lines, triangles, polygons, and bearings develops accurate measurement and multi-step spatial reasoning.

scheduleWhy now? Following right-angled trigonometry, we consolidate triangle and bearing relationships and broaden angle reasoning to parallel lines and polygons. This also supports the accurate drawing, triangle constructions, and angle bisectors encountered later in Year 11.

An acute angle is greater than `0^\circ` and less than `90^\circ`.
A right angle is exactly `90^\circ`.
An obtuse angle is greater than `90^\circ` and less than `180^\circ`.
A reflex angle is greater than `180^\circ` and less than `360^\circ`.
Angles at a point on a straight line add to `180^\circ`.
Angles around a point add to `360^\circ`.
Vertically opposite angles are equal because they are formed by two intersecting straight lines.
The interior angles in a triangle add to `180^\circ`.
An isosceles triangle has two equal sides and the two base angles opposite those equal sides are equal.
The sum of the interior angles in an `n`-sided polygon is `(n-2) \times 180^\circ`.
The sum of the exterior angles of any polygon, taking one exterior angle at each vertex, is `360^\circ`.
A regular polygon has all sides equal and all interior angles equal.
Each interior angle of a regular `n`-sided polygon is `\frac{(n-2) \times 180^\circ}{n}`.
Each exterior angle of a regular `n`-sided polygon is `\frac{360^\circ}{n}`.
Corresponding angles in parallel lines are equal and form matching positions on the two parallel lines.
Alternate angles in parallel lines are equal and form a Z-shape between the two parallel lines.
Co-interior angles in parallel lines add to `180^\circ` and form a C-shape between the two parallel lines.
Bearings are measured clockwise from north and are written as three figures, such as `047^\circ` or `125^\circ`.

rocket_launchYou must be able to

Measure an angle accurately using a protractor by placing the centre on the vertex, lining up the baseline with one arm, and reading the correct scale.
Draw an angle of a given size using a ruler and protractor, marking the required degree measure from a baseline and joining the mark to the vertex.
Estimate the size of an angle by comparing it with key angles such as `90^\circ`, `180^\circ`, `270^\circ`, and `360^\circ`.
Calculate missing angles in right angles, straight lines, around a point, triangles, and polygons by choosing the correct angle sum and subtracting known angles.
Find missing angles using vertically opposite angles by identifying the intersecting straight lines and stating that the opposite angles are equal.
Calculate unknown interior or exterior angles in regular polygons using the number of sides and the correct formula.
Identify and calculate corresponding, alternate, and co-interior angles in parallel lines, giving the correct angle reason for each step.
Solve multi-step angle problems by combining angle facts, including triangle sums, isosceles triangle base angles, angles on a straight line, angles around a point, and parallel line angle facts.
Measure and draw bearings accurately by starting from north, measuring clockwise, and recording the bearing as a three-figure angle.

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