Polygons are multi-sided shapes with different properties. Shapes have symmetrical properties and some can tessellate.

Types of angle

There are 360° in a full turn, 180° in a half turn and 90° in a quarter turn. A quarter turn is called a right angle.

A right angle is shown by a small square. This fact can be used to calculate unknown angles.

Angles at a point and on a straight line

Angles at a point

Angles around a point add up to 360°. This fact can be used to calculate missing angles.

Angles on a straight line

Angles on a straight line add up to 180°. This fact can also be used to calculate angles.

Angles in parallel lines

An introduction to alternate, corresponding and co-interior angles in parallel lines

Parallel lines are lines which are always the same distance apart and never meet. Arrowheads show lines are parallel.

When a pair of parallel lines is cut with another line known as an intersecting transversal, it creates pairs of angles with special properties.

Corresponding angles

Corresponding angles are equal. The lines make an F shape. Notice that the F shape can be upside down or back to front.

Alternate angles

Alternate angles are equal. The lines make a Z shape which can also be back to front.

Vertically opposite angles

Vertically opposite angles are equal.

Triangles

Types of triangle

A triangle is a 2D shape with three sides. There are four different triangles with different properties.

A scalene triangle has 3 sides of different lengths and 3 unequal angles.

An isosceles triangle has 2 sides of equal length. The dashes on the lines show they are equal in length. The angles at the base of the equal sides are equal.

An equilateral triangle has 3 sides of equal length. The dashes on the lines show they are equal in length. All of the angles are also equal.

A right-angled triangle is a triangle that has a right angle.

Labelling angles and sides

Letters can be used to label angles.

AB and AC are line segments, and they meet at point A. AB joins the points A and B.

The angle between AB and AC is labelled BAC.

The angle can be written as BAC or BÂC or ∠BAC.

Interior and exterior angles

The angles inside a shape are called interior angles.

If the side of a triangle is extended, the angle formed outside the triangle is the exterior angle.

Quadrilaterals

A quadrilateral is a 2D shape with four sides.

There are six special quadrilaterals with different properties.

Square

• A square has four sides of equal length.
• It has four right angles (90°).
• The opposite sides are parallel.
• The diagonals bisect each other at right angles.

Rectangle

• A rectangle has two pairs of equal sides.
• It has four right angles (90°).
• The opposite sides are parallel.
• The diagonals bisect each other.

Parallelogram

• A parallelogram has two pairs of equal sides.
• It has two pairs of equal angles.
• The opposite sides are parallel.
• The diagonals bisect each other.

Rhombus

• A rhombus has four sides of equal lengths.
• It has two pairs of equal angles.
• The opposite sides are parallel.
• The diagonals bisect each other at right angles.

Trapezium

• A trapezium has one pair of parallel sides.

Kite

• A kite has two pairs of equal sides.
• It has one pair of equal angles.
• The diagonals bisect at right angles.

Polygons

A polygon is a 2D shape with at least three sides.

Types of polygon

Polygons can be regular or irregular. If the angles are all equal and all the sides are equal length it is a regular polygon.

Interior angles of polygons

To find the sum of interior angles in a polygon divide the polygon into triangles.

The sum of interior angles in a triangle is 180°. To find the sum of interior angles of a polygon, multiply the number of triangles in the polygon by 180°.

Exterior angles of polygons

If the side of a polygon is extended, the angle formed outside the polygon is the exterior angle.

The sum of the exterior angles of a polygon is 360°.

Symmetry

Lines of symmetry

A shape can be folded to see if it has a line of symmetry.

A shape has a line of symmetry when the folded part sits perfectly on top with all edges matching.

Line symmetry in regular polygons

A square is a regular polygon. It has four lines of symmetry and four sides.

A regular pentagon has 5 sides and 5 lines of symmetry.

Rotational symmetry

A shape has rotational symmetry when it can be rotated and it still looks the same.

The order of rotational symmetry of a shape is the number of times it can be rotated around a full circle and still look the same.

If the triangle is rotated a full 360°, it never looks the same except when it arrives back at its original starting position. It only has one order of rotational symmetry, the starting position.

The minimum order of rotational symmetry a shape can have is 1.

Rotational symmetry in regular polygons

The order of rotational symmetry of a regular polygon is equal to the number of sides.

Symmetry of quadrilaterals

There are six special quadrilaterals with different symmetrical properties.

Square

• A square has four lines of symmetry.
• It has rotational symmetry of order four.

Rectangle

• A rectangle has two lines of symmetry.
• It has rotational symmetry of order two.

Parallelogram

• A parallelogram has no lines of symmetry.
• It has rotational symmetry of order two.

Rhombus

• A rhombus has two lines of symmetry.
• It has rotational symmetry of order two.

Trapezium

• Some trapeziums have one line of symmetry. They are called isosceles trapeziums as they have 2 sides of an equal length like isosceles triangles.
• A trapezium has rotational symmetry of order one.

Kite

• A kite has one line of symmetry.
• It has rotational symmetry of order one.