Properties of Quadrilaterals Revision Notes for Leaving Cert Mathematics

Properties of Quadrilaterals

Overview

A quadrilateral is a four-sided polygon with various properties based on its types and classifications. Quadrilaterals include specific types like parallelograms, rectangles, squares, rhombuses, and trapeziums, each having unique properties.

General Properties of Quadrilaterals

Quadrilaterals have four sides, four vertices, and four angles.
The sum of the interior angles of any quadrilateral is $360^\circ$
Opposite sides or angles can have specific relationships depending on the type of quadrilateral.

Types of Quadrilaterals and Their Properties

Parallelogram

Opposite sides are equal and parallel.
Opposite angles are equal.
Diagonals bisect each other.
Consecutive angles are supplementary.

Rectangle

All angles are $90^\circ$
Opposite sides are equal and parallel.
Diagonals are equal and bisect each other.

Rhombus

All sides are equal.
Opposite sides are parallel.
Opposite angles are equal.
Diagonals bisect each other at $90^\circ$
Diagonals bisect the angles of the rhombus.

Square

All properties of a rectangle and a rhombus.
All sides are equal.
All angles are $90^\circ$
Diagonals are equal, bisect each other, and intersect at $90^\circ$

Trapezium (Trapezoid)

Only one pair of opposite sides is parallel (called the bases).
The non-parallel sides are called the legs.
If the legs are equal, it is called an isosceles trapezium.

Kite

Two pairs of adjacent sides are equal.
Diagonals intersect at $90^\circ$
One diagonal bisects the other.

Worked Examples

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Example 1: Diagonals of a Parallelogram

Problem: In a parallelogram, one diagonal measures $10$ cm and divides the parallelogram into two congruent triangles.

Find the length of the other diagonal if the area of the parallelogram is $50$ cm².

Solution:

Step 1: Identify the formula

Area of a parallelogram $= \frac{1}{2} \times d_1 \times d_2$ , where $d_1$ and $d_2$ are diagonals.

Step 2: Given $d_1 = 10$ cm and Area = 50 cm²:

50 = \frac{1}{2} \times 10 \times d_2

Step 3: Solve for $d_2$

d_2 = \frac{50 \times 2}{10} = 10 \, \text{cm}

Answer: The other diagonal is $10$ cm.

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Example 2: Identifying a Quadrilateral

Problem: A quadrilateral has equal diagonals and all angles equal to $90^\circ$ .

Identify the type of quadrilateral.

Solution:

All angles are $90^\circ$ , and diagonals are equal.
This matches the properties of a rectangle.

Answer: The quadrilateral is a rectangle.

Summary

Quadrilaterals are classified into types based on side and angle properties, including parallelograms, rectangles, rhombuses, squares, trapeziums, and kites.
Key properties include relationships between sides, angles, and diagonals.
The sum of the interior angles of any quadrilateral is always $360^\circ$

Properties of Quadrilaterals (Leaving Cert Mathematics): Revision Notes