A quadrilateral is a closed plane figure
bounded by four line segments. E.g. The figure ABCD shown
here is a quadrilateral.
A line segment drawn from one vertex of a quadrilateral to the opposite
vertex is called a diagonal of the quadrilateral. For example, AC is a
diagonal of quadrilateral ABCD.
Types of Quadrilaterals
There are six basic types of quadrilaterals:
1. Rectangle
Opposite sides of a rectangle are parallel and equal. All angles
are 90º.
2. Square
Opposite sides of a square are parallel and all sides are equal.
All angles are 90º.
3. Parallelogram
Opposite sides of a parallelogram are parallel and equal.
Opposite angles are equal.
4. Rhombus
All sides of a rhombus are equal and opposite sides are parallel.
Opposite angles of a rhombus are equal. The diagonals of a rhombus
bisect each other at right angles.
5. Trapezium
A trapezium has one pair of opposite sides parallel. A regular
trapezium has non-parallel sides equal and its base angles are equal, as
shown in the following diagram.
6. Kite
Two pairs of adjacent sides of a kite are equal, and one pair of
opposite angles are equal. Diagonals intersect at right
angles. One diagonal is bisected by the other.
Theorem 3
Prove that the angle sum of a quadrilateral is equal to 360º.
Proof:
Hence the angle sum of a quadrilateral is 360º.
Applying Properties of Angles in Quadrilaterals
The theorems we have proved can be used to prove other theorems.
They can also be used to find the values of the pronumerals in a problem.
Example 13
Find the value of the pronumeral x in the following
diagram. Give reasons for your answer.
Solution:
Example 14
Find the value of the pronumeral x in the following
diagram. Give reasons for your answer.
Solution:
Example 15
Find the value of each of the pronumerals in the following diagram.
Justify your answers.
Solution:
AD = BC, so the trapezium is regular.
Therefore, the base angles are equal.
Example 16
Find the value of each of the pronumerals in the following diagram.
Justify your answers.
Solution:
So x = 15, y = 20.
Example 17
Find the value of each of the pronumerals in the following diagram.
Justify your answers.
Solution:
Example 18
Find the value of each of the pronumerals in
the following diagram.
Solution:
By Pythagoras' theorem from triangle ABD, we have:
Example 19
Find the value of the pronumeral x in the following diagram.
Solution:
By Pythagoras' theorem from triangle ABD, we have:
