As we have seen in Types of Quadrilaterals there are specific geometric properties relating to sides and angles that explicitly define certain shapes.

With relation to all quadrilaterals:

Sum of exterior angles of a polygon is $360$360°
Angle sum of a quadrilateral is $360$360°

Some quadrilaterals have specific angle properties:

Opposite angles in a parallelogram are equal
Angles in a rectangle are equal to 90°
All angles in a square are equal to 90°
Opposite angles of a rhombus are equal
1 pair of opposite equal angles
Standard trapezium - no angle properties

Isosceles trapezium

An isosceles trapezium is a special trapezium where 2 sides are the same length.

It looks like this.

one pair of opposite parallel sides

an isosceles trapezium (trapezoid) has one pair of opposite sides equal

an isosceles trapezium (trapezoid) has 2 pairs of adjacent angles equal

ANGLES FORMED BY Diagonals in quadrilaterals

In addition to the properties already studied, the angles formed by the diagonals of some quadrilaterals also have special properties.

Diagonals of a square bisect the angles at the vertices (makes them 45°)

Diagonals of a square are perpendicular to each other (cross at 90°)

Diagonals of a rhombus bisect corner angles.Diagonals of a rhombus bisect each other at 90 degrees ($BO=DO$BO=DO and $AO=CO$AO=CO)

i.e. $\angle OAB=\angle OAD$∠OAB=∠OAD, $\angle OCD=\angle OCB$∠OCD=∠OCB,

$\angle OBC=\angle OBA$∠OBC=∠OBA and $\angle ODC=\angle ODA$∠ODC=∠ODA

i.e. $\angle BAO=\angle DAO$∠BAO=∠DAO ( b=a)

The longest diagonal of a kite bisects the angles through which it passes.

$\angle COD=\angle COB$∠COD=∠COB , i.e. (m=n)

Diagonals of a kite are perpendicular to each other.

The following applet will allow you to manipulate different quadrilaterals using the blue points and see the properties appear with regards to the diagonals.

Created with Geogebra

Worked Examples

Question 1

Find the value of $z$z in the following figure.

Question 2

Given the following kite ABCD, calculate $x$x

Question 3

Find the value of all variables in the figure, giving reasons.

a) Find $x$x

b) Find $y$y

Angles in Quadrilaterals