Properties of ShapesAQA GCSE Maths: Revision notes
Section 1
What angle properties exist at a point and on a line?
Angles at a point sum to 360°. Angles on a straight line sum to 180°. Vertically opposite angles (formed when two straight lines cross) are always equal.
When working with parallel lines cut by a transversal (a line crossing them):
- Alternate angles are equal (on opposite sides of the transversal, between the parallel lines)
- Corresponding angles are equal (on the same side of the transversal, one interior and one exterior)
- Co-interior angles (also called consecutive interior or same-side interior angles) sum to 180° (both between the parallel lines, on the same side of the transversal)
| Angle Type | Position | Property |
|---|---|---|
| Alternate | Opposite sides of transversal, between parallels | Equal |
| Corresponding | Same side of transversal, one in/one out | Equal |
| Co-interior | Same side of transversal, both between | Sum to 180° |
| Vertically opposite | Opposite sides of intersection | Equal |
When answering angle questions, always state which property you are using (e.g., 'angles on a straight line' or 'alternate angles with parallel lines'). Examiners expect to see the reasoning, not just the answer.
Students often confuse alternate and corresponding angles. Remember: alternate angles are on opposite sides of the transversal and are between the parallel lines; corresponding angles are on the same side.
Section 2
How do triangle and polygon angle properties work?
The sum of angles in a triangle is always 180°. This is a fundamental fact you must know.
For polygons with more sides, use the formula: Sum of interior angles = (n − 2) × 180°, where n is the number of sides.
| Polygon | Number of sides | Sum of interior angles |
|---|---|---|
| Triangle | 3 | 180° |
| Quadrilateral | 4 | 360° |
| Pentagon | 5 | 540° |
| Hexagon | 6 | 720° |
| Heptagon | 7 | 900° |
| Octagon | 8 | 1080° |
For a regular polygon (all sides and angles equal), divide the sum by the number of sides to find each interior angle: Interior angle = (n − 2) × 180° ÷ n
Exterior angles of any polygon sum to 360°. For a regular polygon, each exterior angle = 360° ÷ n. Note that interior angle + exterior angle = 180° (they form a straight line at each vertex).
Find each interior angle of a regular hexagon. Use (n − 2) × 180° ÷ n with n = 6: (6 − 2) × 180° ÷ 6 = 4 × 180° ÷ 6 = 720° ÷ 6 = 120°.
For any polygon, exterior angles always sum to 360° regardless of the number of sides. This is quicker to use than interior angles in some problems.
Section 3
What are the key properties of triangles and quadrilaterals?
Triangle types and their properties:
| Triangle Type | Properties |
|---|---|
| Equilateral | All three sides equal; all angles = 60°; has 3 lines of symmetry |
| Isosceles | Two sides equal; two angles equal (the base angles); 1 line of symmetry |
| Scalene | All sides different lengths; all angles different |
| Right-angled | One angle = 90°; can be isosceles or scalene; Pythagoras' theorem applies |
Quadrilateral types and their properties:
| Quadrilateral | Properties |
|---|---|
| Square | All sides equal; all angles 90°; diagonals equal and bisect at 90°; 4 lines of symmetry |
| Rectangle | Opposite sides equal and parallel; all angles 90°; diagonals equal and bisect; 2 lines of symmetry |
| Parallelogram | Opposite sides equal and parallel; opposite angles equal; diagonals bisect each other (not at 90°) |
| Rhombus | All sides equal; opposite sides parallel; opposite angles equal; diagonals bisect at 90°; 2 lines of symmetry |
| Trapezium | One pair of parallel sides; no line symmetry (unless isosceles trapezium) |
| Kite | Two pairs of adjacent equal sides; one line of symmetry; diagonals perpendicular |
All quadrilaterals have interior angles summing to 360°.
When identifying or describing shapes, always mention the properties that make them special (equal sides, parallel sides, angle sizes, symmetry). Examiners want to see you understand why a shape is classified as it is.
Think of quadrilaterals as a family: a square is the most 'perfect' member (all sides equal, all angles 90°), a rectangle relaxes one rule (only opposite sides equal), a parallelogram relaxes another (angles don't have to be 90°), and a trapezium has just one rule (one pair of parallel sides).
Section 4
What are circle properties and how do circle theorems work?
Key circle definitions:
- Radius: distance from centre to the circumference
- Diameter: a straight line through the centre joining two points on the circumference (diameter = 2 × radius)
- Chord: a straight line joining two points on the circumference (not necessarily through the centre)
- Arc: a curved part of the circumference
- Sector: the region between two radii and an arc (like a slice of pie)
- Segment: the region between a chord and the arc it cuts off
- Tangent: a straight line that touches the circumference at exactly one point
Essential circle theorems:
-
Angle at centre is twice the angle at circumference: When two angles are subtended by the same arc, the angle at the centre is twice the angle at the circumference.
-
Angles in the same segment are equal: All angles subtended by the same arc from points on the circumference are equal.
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Opposite angles in a cyclic quadrilateral sum to 180°: A cyclic quadrilateral is one inscribed in a circle; both pairs of opposite angles are supplementary.
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Tangent–radius relationship: The tangent to a circle is perpendicular (at 90°) to the radius at the point of contact.
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Alternate segment theorem (Higher Tier): The angle between a tangent and a chord equals the angle in the alternate segment (the angle subtended by the chord from the opposite side of the circle).
An angle at the circumference is 35°. Find the angle at the centre subtended by the same arc. Using the theorem: angle at centre = 2 × angle at circumference = 2 × 35° = 70°.
Always identify which circle theorem applies before solving. State it clearly in your working (e.g., 'angles in the same segment are equal' or 'opposite angles in a cyclic quadrilateral'). For Higher Tier, be prepared to prove theorems using angle properties.
Students often forget that the tangent is perpendicular to the radius. This 90° angle is crucial in many circle problems and is frequently tested.
Section 5
How do you calculate interior and exterior angles of regular and irregular polygons?
For regular polygons (all sides and angles equal):
- Interior angle = (n − 2) × 180° ÷ n, where n = number of sides
- Exterior angle = 360° ÷ n
- Relationship: interior angle + exterior angle = 180°
For irregular polygons (sides and/or angles not all equal):
- Sum of all interior angles = (n − 2) × 180°
- If you know all but one interior angle, subtract the known angles from the total to find the missing one
- Sum of exterior angles = 360° (for any polygon)
- Each exterior angle is supplementary to its corresponding interior angle (they sum to 180°)
Step-by-step approach for irregular polygons:
- Identify the number of sides (n)
- Calculate the sum of interior angles: (n − 2) × 180°
- Add up the angles you know
- Subtract from the total to find any unknown angles
- To find an exterior angle, subtract the interior angle from 180°
Key fact: The exterior angle of any polygon at any vertex equals 180° minus the interior angle at that vertex.
A pentagon has interior angles of 120°, 110°, 130°, and 125°. Find the fifth angle. Sum of interior angles in a pentagon = (5 − 2) × 180° = 540°. Fifth angle = 540° − 120° − 110° − 130° − 125° = 55°.
When a question gives you exterior angles, remember they always sum to 360°. This can be a quick check: add up all exterior angles given and subtract from 360° to find any missing exterior angle.
Must Know
- Angles on a straight line sum to 180°; angles at a point sum to 360°; vertically opposite angles are equal
- Alternate and corresponding angles are equal when a transversal crosses parallel lines; co-interior angles sum to 180°
- Sum of angles in a triangle = 180°; sum of interior angles in a polygon = (n − 2) × 180°
- For regular polygons: interior angle = (n − 2) × 180° ÷ n and exterior angle = 360° ÷ n; exterior angles of any polygon always sum to 360°
- Know the properties of all triangle types (equilateral, isosceles, scalene, right-angled) and quadrilateral types (square, rectangle, parallelogram, rhombus, trapezium, kite)
- Circle theorems: angle at centre = 2 × angle at circumference; angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to 180°; tangent is perpendicular to radius; alternate segment theorem (HT)
- In any polygon, interior angle + exterior angle = 180° at each vertex
That's the notes covered.
Carry on to the next subtopic.