Geometry Toolkit

Geometry

Types of triangle

Equilateral: three equal sides and three angles of 60°, with three lines of symmetry
Isosceles: two equal sides and the two angles opposite them equal, with one line of symmetry
Scalene: all three sides different and all three angles different, with no line of symmetry
Right-angled: contains one angle of 90°, and it may also be isosceles or scalene
A question giving the sides as equal expects the name, so mark equal sides on the diagram before deciding

The four triangle types with their angle facts: equilateral with three dashed equal sides and three 60 degree angles, isosceles with two equal sides and two equal base angles, scalene with all three angles different, and right-angled where the two remaining angles add to 90 — Source: Properties of 2D shapes by Save My Exams

The six special quadrilaterals

Shape	Sides	Angles	Diagonals	Symmetry
Square	4 equal, 2 pairs parallel	all 90°	bisect each other, equal, perpendicular	4 lines, order 4
Rectangle	2 pairs equal and parallel	all 90°	bisect each other, equal	2 lines, order 2
Rhombus	4 equal, 2 pairs parallel	opposite angles equal	bisect each other, perpendicular	2 lines, order 2
Parallelogram	2 pairs equal and parallel	opposite angles equal	bisect each other	no lines, order 2
Trapezium	exactly one pair parallel	two pairs adding to 180°	do not bisect	none, order 1
Kite	2 pairs of equal adjacent sides	one pair of opposite angles equal	one bisects the other, perpendicular	1 line, order 1

A square is a rectangle with all sides equal, and a rhombus is a parallelogram with all sides equal

The six special quadrilaterals drawn with their side markings: square, rectangle, parallelogram, rhombus, trapezium and kite, each dashed to show which sides are equal — Source: Properties of 2D shapes by Save My Exams

An isosceles trapezium adds one line of symmetry, equal non-parallel sides and equal diagonals

A plain trapezium with one pair of parallel sides and its diagonals drawn dashed, beside an isosceles trapezium ABCD where the two base angles at A and B are equal, the two top angles at C and D are equal, and the diagonals are equal — Source: Properties of 2D shapes by Save My Exams

Questions describe a shape by its properties and ask for its name, so read every property before answering
- One line of symmetry and diagonals crossing at right angles describes a kite
- Four equal sides, two lines of symmetry and rotational symmetry of order 2 describes a rhombus, since a square would have four lines

A kite ABCD with the two short sides AB and AD marked equal and the two long sides CB and CD marked equal, and its diagonals crossing at right angles at O, the long diagonal AC cutting the short one in half — Source: Properties of 2D shapes by Save My Exams

A rectangle ABCD with both diagonals drawn meeting at E: the two diagonals are marked equal to each other and each is cut exactly in half at E, and all four corners are right angles — Source: Properties of 2D shapes by Save My Exams

A parallelogram ABCD with arrows showing AB parallel to DC and AD parallel to BC, and both diagonals drawn meeting at E, dashes showing each diagonal is cut exactly in half there — Source: Properties of 2D shapes by Save My Exams

Worked example

Naming a quadrilateral from its properties

In a quadrilateral the equal sides occur in two pairs that meet at a corner, no side is parallel to any other, and there is exactly one line of symmetry. Write down its mathematical name.

Solution:

Work through the six special quadrilaterals against each property in turn
"No parallel sides" rules out the square, rectangle, rhombus, parallelogram and trapezium, since every one of those has at least one parallel pair
The remaining shape is the kite, so check the other two properties against it
A kite is built from equal sides that sit next to one another rather than opposite, which matches
A kite has exactly one line of symmetry, along the longer diagonal, which also matches
The quadrilateral is a kite