Properties of Triangles & Quadrilaterals

Quick revise

After studying this section, you will be able to:

  • use the angle sum of a triangle and a quadrilateral
  • identify quadrilaterals by their geometric properties

The video below looks at the properties of triangles and quadrilaterals.

Sum of angles of a triangle

You need to be able to prove that:

(a) the sum of the angles in a triangle is 180°.

Take any triangle ABC. Construct XY through B and parallel to AC. Using the properties of parallel
lines angle A = angle XBA and angle C = angle CBY. Hence the angle sum of the triangle is angle A + angle ABC + angle C = angle XBA + angle ABC + angle CBY = 180° = angles on a straight line.

You must remember the basic angle facts such as the sum of the angles on a straight line is 180°, and the properties of alternate and corresponding angles.

sumof angles in a triangle

(b) the exterior angle of a triangle is equal to the sum of the interior opposite angles.

Take any triangle ABC. Construct a line through C, parallel to AB.

angle p = angle b (corresponding angles)
angle s = angle a (alternate angles)

Therefore angle p + angle s = angle a + angle b

but angle r = angle a + angle b

Therefore angle p + angle s = angle r

Sum of angles in a triangle

Properties of quadrilaterals

Sum of angles in a quadrilateral

You can use the fact that the sum of the angles in a triangle = 180° to prove that the angle sum of a quadrilateral is 360°.

angles a + b + c = 180°

angles d + e + f = 180°

Therefore a + b + c + d + e + f = 360°

properties of a quadrilateral

Geometric properties of quadrilaterals

You need to be able to identify quadrilaterals by their geometric properties.

(a) Square

  • all sides equal and opposite sides parallel
  • all angles 90°
  • four lines of symmetry
  • rotational symmetry order 4
  • diagonals bisect at right angles

square

(b) Rectangle

  • opposite sides equal and parallel
  • all angles 90°
  • two lines of symmetry
  • rotational symmetry order 2

rectangle

(c) Parallelogram

  • opposite sides equal and parallel
  • opposite angles equal
  • no lines of symmetry
  • rotational symmetry order 2

 parallelogram

(d) Rhombus

  • all sides equal
  • opposite sides parallel
  • two lines of symmetry
  • rotational symmetry order 2
  • diagonals bisect at right angles

 rhombus

(e) Kite

  • one line of symmetry
  • diagonals intersect at right angles

 kite

(f) Trapezium

  • one pair of sides parallel

 trapezium

examples