Properties of 2D Shapes

For Edexcel GCSE Mathematics, you need to know the properties of triangles, quadrilaterals and circles in detail. This means knowing not just the names, but the exact properties of sides, angles, diagonals and symmetry. This lesson provides a comprehensive reference and worked examples.

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Key Vocabulary

Term	Meaning
Diagonal	A line joining two non-adjacent vertices of a polygon
Line of symmetry	A line that divides a shape into two mirror-image halves
Rotational symmetry	The shape looks the same after a rotation of less than 360°; the order tells you how many times
Perpendicular	At right angles (90°) to each other
Bisect	To cut exactly in half

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Triangles

Triangle type	Sides	Angles	Lines of symmetry	Rotational symmetry order
Equilateral	All 3 equal	All 60°	3	3
Isosceles	2 equal	2 equal (base angles)	1	1
Scalene	All different	All different	0	1
Right-angled	Varies	One angle = 90°	0 (unless isosceles right-angled: 1)	1

Worked Example 1

Triangle ABC has AB = AC = 7 cm and angle BAC = 50°. Name the type of triangle and find angles ABC and ACB.

Since AB = AC, the triangle is isosceles. The base angles are equal. 50 + 2b = 180, so 2b = 130, b = 65. Angle ABC = angle ACB = 65°.

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Quadrilaterals

Quadrilateral	Sides	Angles	Diagonals	Lines of symmetry	Rot. sym. order
Square	All 4 equal	All 90°	Equal, bisect at 90°	4	4
Rectangle	Opposite pairs equal	All 90°	Equal, bisect each other (not at 90°)	2	2
Parallelogram	Opposite pairs equal	Opposite angles equal; co-interior pairs sum to 180°	Bisect each other (not equal, not at 90°)	0	2
Rhombus	All 4 equal	Opposite angles equal	Bisect at 90° (not equal in length)	2	2
Trapezium	One pair of parallel sides	Varies	Not equal in general	0 (isosceles trapezium: 1)	1
Isosceles trapezium	One pair parallel, non-parallel sides equal	Two pairs of equal angles	Equal in length	1	1
Kite	Two pairs of adjacent sides equal	One pair of opposite angles equal (between unequal sides)	One bisects the other at 90°	1	1

Exam Tip: Questions often give you a quadrilateral and ask you to identify it from its properties, or ask you to state properties of a named quadrilateral. Learn the diagonal properties — these are a common source of marks.

Worked Example 2

A quadrilateral has all four sides equal and diagonals that bisect each other at right angles, but the angles are not all 90°. Name the shape.

The shape is a rhombus.

Worked Example 3

PQRS is a parallelogram. Angle P = 65°. Find all the other angles.

In a parallelogram, opposite angles are equal and co-interior angles sum to 180°. Angle R = 65° (opposite to P). Angle Q = 180 - 65 = 115° (co-interior with P). Angle S = 115° (opposite to Q).

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Parts of a Circle

You must know the following circle terminology for Edexcel GCSE:

Term	Definition
Centre	The fixed point equidistant from all points on the circle
Radius	A line from the centre to the circumference
Diameter	A line through the centre, from one side of the circumference to the other (= 2 x radius)
Circumference	The perimeter (outer edge) of the circle
Chord	A straight line joining two points on the circumference
Arc	A portion of the circumference
Sector	The region between two radii and an arc (a "pizza slice")
Segment	The region between a chord and an arc
Tangent	A straight line that touches the circumference at exactly one point and is perpendicular to the radius at that point

Important: A tangent to a circle is perpendicular to the radius at the point of contact. This fact is used in Higher tier circle theorem questions.

Worked Example 4

Identify the shaded region: (a) The region between two radii and the major arc is the major sector. (b) The region between a chord and the minor arc is the minor segment.

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Circle Theorems (Higher) [H]

Edexcel Higher tier students must also know:

1. The angle at the centre is twice the angle at the circumference (subtended by the same arc).
2. Angles in the same segment are equal.
3. The angle in a semicircle is 90°.
4. Opposite angles of a cyclic quadrilateral sum to 180°.
5. A tangent is perpendicular to the radius at the point of contact.
6. Two tangents from an external point are equal in length.
7. The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

Worked Example 5 [H]

O is the centre of a circle. Angle AOB = 124° where A and B are points on the circumference. Find the angle ACB where C is another point on the circumference on the major arc side.

Angle at the centre = 2 x angle at circumference. 124 = 2 x angle ACB Angle ACB = 124 / 2 = 62°

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Special Quadrilateral Relationships

It helps to understand how the quadrilaterals relate to each other:

• A square is a special rectangle (all sides equal) and a special rhombus (all angles 90°).
• A rectangle is a special parallelogram (all angles 90°).
• A rhombus is a special parallelogram (all sides equal).
• A parallelogram is a special trapezium (both pairs of opposite sides parallel).

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Common Mistakes and Misconceptions