Properties of Quadrilaterals (Leaving Cert Mathematics): Revision Notes
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
- 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
- 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
- Diagonals bisect the angles of the rhombus.
Square
- All properties of a rectangle and a rhombus.
- All sides are equal.
- All angles are
- Diagonals are equal, bisect each other, and intersect at
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
- One diagonal bisects the other.
Worked Examples
Example 1: Diagonals of a Parallelogram
Problem: In a parallelogram, one diagonal measures cm and divides the parallelogram into two congruent triangles.
Find the length of the other diagonal if the area of the parallelogram is cm².
Solution:
Step 1: Identify the formula
Area of a parallelogram , where and are diagonals.
Step 2: Given cm and Area = 50 cm²:
Step 3: Solve for
Answer: The other diagonal is cm.
Example 2: Identifying a Quadrilateral
Problem: A quadrilateral has equal diagonals and all angles equal to .
Identify the type of quadrilateral.
Solution:
- All angles are , 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