Quadrilaterals

Quadrilaterals which are closed figures with 4 sides, are the next simplest shapes after triangles. Unlike triangles, quadrilaterals are not rigid structures.

Two of the fundamental properties of quadrilaterals in the Angle Sum property and the midpoints of the sides forming a parallelogram.

Angle Sum

Any quadrilateral can be divided into 2 triangles by drawing any of the two diagonals. Hence the total of the interior angles of a quadrilateral is the sum of the interior angles of the 2 triangles into which it has been divided, which is 4 right angles or 360 degrees.

Midpoints of a Quadrilateral form a Parallelogram

Imagine a quadrilateral as made of 2 triangles with one of the diagonals being the common base. If we join the mid points of the other 2 sides of the triangles, the line joining them would be parallel to the base of the triangle. The same hold true for the line joining the mid points of the second triangle. Since the base of both the triangles is the same, both the lines are also parallel.

By separating the 2 triangles by drawing the other diagonal, we can prove that the other 2 sides of the figure formed by joining the also parallel. Hence the figure will always be a parallelogram.

Types of Quadrilaterals

Convex & Concave Quadrilaterals

All quadrilaterals can be classified into concave & convex. In a concave quadrilateral, one of the internal angles will be a reflex angle (>180). They roughly resemble the tip of an arrow. They are not studied in detail in school geometry.

In a convex quadrilateral all the internal angles are less than a straight angle.

Types of convex quadrilaterals

There are several well-known types of convex quadrilaterals. A major classification of quadrilaterals is Trapezium & Kite.

Properties & relations of various types of quadrilaterals. Chapter 23.10 give a visual representation which would make understanding easier and make memorization unnecessary.

Trapezium - If any 2 opposing sides of the quadrilateral are parallel, it is a Trapezium. Trapezium was studied well in ancient cultures. It resembles the cross section of a canal. All cultures depended upon water to transport heavy loads and had to dig canals. The volume of earth to be dug out depended on its cross-sectional area. Hence the formula for the area of a trapezium was worked out in ancient cultures.

Kite - If on the other hand, in a quadrilateral, both pairs of adjacent sides are equal, it is a Kite. The diagonals of a Kite are perpendicular to each other. A Kite can be divided by one of the diagonals into 2 isosceles triangles, which have the same 3rd(non-equal) side. The other diagonal divides a Kite into 2 congruent triangles.

If the other 2 sides of the Trapezium are also parallel then we get a Parallelogram. We will study Parallelograms and shapes related to them in the next chapter.

Rhombus - In the next chapter, we will also come across the Rhombus, which belongs both the Parallelogram & Kite families. If in a Kite, the opposing sides are equal, then also we get a Rhombus.

Chapter 23.10 is arranged like a family tree, giving the relations between various kinds of quadrilaterals in a logical manner.

We can say a Rhombus is a Kite which is a Quadrilateral. We can also say a Rhombus is a Parallelogram which is a Trapezium which is a Quadrilateral.

Area - The area of a quadrilateral can be measured by dividing it into 2 triangles whose area can be worked out individually. In various types of quadrilaterals, because of their symmetry, simple formulas can be worked out for finding the area.

Cyclic Quadrilaterals

If a circle can be drawn through the four vertices of a quadrilateral, it is called a Cyclic Quadrilateral. Or we can say that, if any four points on the circumference of a circle are joined to form a quadrilateral, it would be a cyclic quadrilateral. A cyclic quadrilateral has some special properties.

The opposite angles of a cyclic quadrilateral will be supplementary or total to 180 degrees.

Conversely, we can say that if the opposite angles of a quadrilateral add up to 180 degrees, then a circle can be drawn passing through all the four vertices. A square and a rectangle are examples of such quadrilaterals.

The proof of this can be understood after we study properties of a circle.