Triangles & Quadrilaterals (SQA National 5 Maths): Revision Note

Angles in triangles and quadrilaterals

What are the angle sums of triangles and quadrilaterals?

• The angles inside any triangle will add up to 180°

• The angles inside any quadrilateral (four-sided shape) will add up to 360°

How can I find a missing angle in a triangle or quadrilateral if I know all the other angles?

• To find a missing angle in a triangle

  • subtract all the other angles from 180°

• To find a missing angle in a quadrilateral

  • subtract all the other angles from 360°

Other properties of triangles and quadrilaterals

You should be familiar with the properties of different types of triangles and quadrilaterals from your National 4 Maths course.

The key properties are summarised below.

What are the names of the different types of triangles?

• You should know the names and properties of the different types of triangles

  • An equilateral triangle has 3 equal sides and 3 equal angles

  • An isosceles triangle has 2 equal sides and 2 equal angles

  • A right-angled triangle has one 90° angle

  • A scalene triangle has 3 sides all of different lengths

What are the names of the different types of quadrilaterals?

• You should know the names and properties of the different types of quadrilaterals

  • These are squares, rectangles, parallelograms, rhombuses, trapeziums and kites

What are the properties of rectangles and squares?

• Rectangles and squares have four equal right angles (90°)

• Rectangles have two pairs of equal length, parallel sides

  • Squares are just regular rectangles; all four of their sides are equal

• The diagonals of a rectangle bisect each other at the centre of the rectangle

  • This means that they cut each other in half

  • The intersecting diagonals form two pairs of angles at the centre

    • In a square, all four of these angles will be equal to 90°

• Pythagoras’ theorem can be used to find the length of the diagonal of a square or rectangle

  • The diagonal forms the hypotenuse of a right-angled triangle

What are the properties of parallelograms and rhombuses?

• Parallelograms and rhombuses (rhombi) have two pairs of equal, opposite, angles

• Parallelograms and rhombuses have two pairs of opposite, parallel sides

• Rhombuses have four sides of the same length

  • This means a rhombus is a regular parallelogram

    • A square is also a regular rhombus

• The diagonals of a parallelogram bisect each other, forming two pairs of opposite angles

• The diagonals of a rhombus bisect each other at right angles (90°)

  • This means that they cut each other in half

  • The diagonals will not be of equal length

    • On the diagram below, the diagonal AC is shorter than the diagonal DB

What are the properties of trapeziums?

• Trapeziums have one pair of opposite, parallel sides

  • These are not of equal length

• Trapeziums may not have any equal angles

  • As with all quadrilaterals, the angles add up to 360°

• If a trapezium has a line of symmetry, it is classed as isosceles

  • Isosceles trapeziums have two pairs of equal angles

  • The non-parallel sides in an isosceles trapezium will be equal length

  • An isosceles trapezium has two diagonals of equal length

What are the properties of kites?

• Kites have one line of symmetry, known as their main diagonal

• The angles opposite the main diagonal are equal

  • These are angles ABC and ADC on the diagram below

• The diagonals of a kite bisect each other at right angles (90°)

  • This means that they cut each other in half

  • The diagonals will not be of equal length

• Kites have no parallel sides

• Kites have two pairs of equal length, adjacent sides