# Properties of Triangles & Quadrilaterals

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.

(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

**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°

**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

(b) Rectangle

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

(c) Parallelogram

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

(d) Rhombus

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

(e) Kite

- one line of symmetry
- diagonals intersect at right angles

(f) Trapezium

- one pair of sides parallel