QUADRILATERAL: ELEMENTS, PROPERTIES, CLASSIFICATION, EXAMPLES - MATHS - 2025

A quadrilateral is a polygon with four sides and four vertices. Its opposite sides are those that do not have vertices in common, while consecutive sides are those that have a common vertex.

In a quadrilateral, adjacent angles share one side, while opposite angles have no sides in common. Another important characteristic of a quadrilateral is that the sum of its four internal angles is twice the plane angle, that is, 360º or 2π radians.

Figure 1. Various quadrilaterals. Source: F. Zapata.

Diagonals are the segments that join a vertex with its opposite and in a given quadrilateral, a single diagonal can be drawn from each vertex. The total number of diagonals in a quadrilateral is two.

Quadrilaterals are figures known to mankind since ancient times. Archaeological records, as well as the constructions that survive today, attest to this.

Likewise, today the quadrilaterals continue to have an important presence in everyone's daily life. The reader can find this form on the screen on which he is reading the text at this very moment, on windows, doors, automotive parts, and countless other places.

Quadrilateral classification

According to the parallelism of the opposite sides, the quadrilaterals are classified as follows:

1. Trapezoid, when there is no parallelism and the quadrilateral is convex.
2. Trapezoid, when there is parallelism between a single pair of opposite sides.
3. Parallelogram, when its opposite sides are parallel two by two.

Figure 2. Classification and subclassification of quadrilaterals. Source: Wikimedia Commons.

Types of parallelogram

In turn, the parallelograms can be classified according to their angles and their sides as follows:

1. Rectangle is the parallelogram that has its four internal angles of equal measure. The interior angles of a rectangle form a right angle (90º).
2. Square, it is a rectangle with its four sides of equal measure.
3. Rhombus is the parallelogram with its four equal sides, but different adjacent angles.
4. Rhomboid, parallelogram with different adjacent angles.

Trapeze

The trapezoid is a convex quadrilateral with two parallel sides.

Figure 3. Bases, sides, height and median of a trapezoid. Source: Wikimedia Commons.

- In a trapezoid, the parallel sides are called bases and the non-parallel sides are called laterals.

- The height of a trapezoid is the distance between the two bases, that is, the length of a segment with ends at the bases and perpendicular to them. This segment is also called a height of the trapezoid.

- The median is the segment that joins the midpoints of the laterals. It can be shown that the median is parallel to the bases of the trapezoid and its length is equal to the semisum of the bases.

- The area of ​​a trapezoid is its height multiplied by the semi-sum of the bases:

Types of trapezoids

-Rectangular trapezoid: it is the one with a side perpendicular to the bases. This side is also the height of the trapezium.

-Isosceles trapezoid: the one with sides of equal length. In an isosceles trapezoid the angles adjacent to the bases are equal.

-Scalene trapezium: the one with its sides of different lengths. Its opposite angles can be one acute and the other obtuse, but it can also happen that both are obtuse or both acute.

Figure 4. Types of trapezium. Source: F. Zapata.

Parallelogram

The parallelogram is a quadrilateral whose opposite sides are parallel two by two. In a parallelogram the opposite angles are equal and the adjacent angles are supplementary, or put another way, the adjacent angles add up to 180º.

If a parallelogram has a right angle, then all other angles will be too, and the resulting figure is called a rectangle. But if the rectangle also has its adjacent sides of the same length, then all its sides are equal and the resulting figure is a square.

Figure 5. Parallelograms. The rectangle, the square, and the rhombus are parallelograms. Source: F. Zapata.

When a parallelogram has two adjacent sides of the same length, all of its sides will be the same length, and the resulting figure is a rhombus.

The height of a parallelogram is a segment with ends on its opposite sides and perpendicular to them.

Area of ​​a parallelogram

The area of ​​a parallelogram is the product of the base times its height, the base being a side perpendicular to the height (figure 6).

Diagonals of a parallelogram

The square of the diagonal that starts from a vertex is equal to the sum of the squares of the two sides adjacent to said vertex plus the double product of those sides by the cosine of the angle of that vertex:

f ² = a ² + d ² + 2 ad Cos (α)

Figure 6. Parallelogram. Opposite angles, height, diagonals. Source: F. Zapata.

The square of the diagonal opposite the vertex of a parallelogram is equal to the sum of the squares of the two sides adjacent to said vertex and subtracting the double product of those sides by the cosine of the angle of that vertex:

g ² = a ² + d ² - 2 ad Cos (α)

Law of parallelograms

In any parallelogram, the sum of the squares of its sides is equal to the sum of the squares of the diagonals:

a ² + b ² + c ² + d ² = f ² + g ²

re ctángulo

The rectangle is a quadrilateral with its opposite sides parallel two by two and which also has a right angle. In other words, the rectangle is a type of parallelogram with a right angle. Because it is a parallelogram, the rectangle has opposite sides of equal length a = c and b = d.

But as in any parallelogram the adjacent angles are supplementary and the opposite angles equal, in the rectangle because it has a right angle, it will necessarily form right angles in the other three angles. In other words, in a rectangle all the internal angles measure 90º or π / 2 radians.

Diagonals of a rectangle

In a rectangle the diagonals are of equal length, as will be demonstrated below. The reasoning is as follows; A rectangle is a parallelogram with all its right angles and therefore inherits all the properties of the parallelogram, including the formula that gives the length of the diagonals:

f ² = a ² + d ² + 2 ad Cos (α)

g ² = a ² + d ² - 2 ad Cos (α)

with α = 90º

Since Cos (90º) = 0, then it happens that:

f ² = g ² = a ² + d ²

That is, f = g, and therefore the lengths f and g of the two diagonals of the rectangle are equal and their length is given by:

Furthermore, if in a rectangle with adjacent sides a and b one side is taken as the base, the other side will be height and consequently the area of ​​the rectangle will be:

Area of ​​the rectangle = ax b.

The perimeter is the sum of all the sides of the rectangle, but since the opposites are equal, it follows that for a rectangle with sides a and b the perimeter is given by the following formula:

Perimeter of rectangle = 2 (a + b)

Figure 7. Rectangle with sides a and b. The diagonals f and g are of equal length. Source: F. Zapata.

Square

The square is a rectangle with its adjacent sides the same length. If the square has side a, then its diagonals f and g have the same length, which is f = g = (√2) a.

The area of ​​a square is its side squared:

Area of ​​a square = a ²

The perimeter of a square is twice the side:

Perimeter of a square = 4 a

Figure 8. Square with side a, indicating its area, its perimeter and the length of its diagonals. Source: F. Zapata..

Diamond

The rhombus is a parallelogram with its adjacent sides the same length, but since in a parallelogram the opposite sides are equal then all the sides of a rhombus are equal in length.

The diagonals of a rhombus are of different length, but they intersect at right angles.

Figure 9. Rhombus of side a, indicating its area, its perimeter and the length of its diagonals. Source: F. Zapata.

Examples

Example 1

Show that in a quadrilateral (not crossed) the internal angles add up to 360º.

Figure 10: It is shown how the sum of the angles of a quadrilateral add up to 360º. Source: F. Zapata.

A quadrilateral ABCD is considered (see figure 10) and the diagonal BD is drawn. Two triangles ABD and BCD are formed. The sum of the interior angles of triangle ABD is:

α + β ₁ + δ ₁ = 180º

And the sum of the internal angles of triangle BCD is:

β2 + γ + δ ₂ = 180º

Adding the two equations we obtain:

α + β ₁ + δ ₁ + β ₂ + γ + δ ₂ = 180º + 180º

Grouping:

α + (β ₁ + β ₂) + (δ ₁ + δ ₂) + γ = 2 * 180º

By grouping and renaming, it is finally shown that:

α + β + δ + γ = 360º

Example 2

Show that the median of a trapezoid is parallel to its bases and its length is the semisum of the bases.

Figure 11. Median MN of the trapezium ABCD. Source: F. Zapata.

The median of a trapezoid is the segment that joins the midpoints of its sides, that is, the non-parallel sides. In the trapezoid ABCD shown in figure 11 the median is MN.

Since M is the midpoint of AD and N is the midpoint of BC, the AM / AD and BN / BC ratios are equal.

That is, AM is proportional to BN in the same proportion as AD is to BC, so the conditions for the application of Thales' (reciprocal) theorem are given, which states the following:

"If proportional segments are determined in three or more lines cut by two secants, then these lines are all parallel."

In our case it is concluded that the lines MN, AB and DC are parallel to each other, therefore:

"The median of a trapezoid is parallel to its bases."

Now the Thales theorem will be applied:

"A set of parallels cut by two or more secants determine proportional segments."

In our case AD = 2 AM, AC = 2 AO, so the triangle DAC is similar to the triangle MAO, and consequently DC = 2 MO.

A similar argument allows us to affirm that CAB is similar to CON, where CA = 2 CO and CB = 2 CN. It immediately follows that AB = 2 ON.

In short, AB = 2 ON and DC = 2 MO. So when adding we have:

AB + DC = 2 ON + 2 MO = 2 (MO + ON) = 2 MN

Finally MN is cleared:

MN = (AB + DC) / 2

And it is concluded that the median of a trapezoid measures the semi-sum of the bases, or put another way: the median measures the sum of the bases, divided by two.

Example 3

Show that in a rhombus the diagonals intersect at right angles.

Figure 12. Rhombus and demonstration that its diagonals intersect at right angles. Source: F. Zapata.

The blackboard in figure 12 shows the necessary construction. First the parallelogram ABCD is drawn with AB = BC, that is, a rhombus. Diagonals AC and DB determine eight angles shown in the figure.

Using the theorem (aip) which states that alternate interior angles between parallels cut by a secant determine equal angles, we can establish the following:

α ₁ = γ ₁, α2 = γ2, δ ₁ = β ₁ and δ2 = β2. (*)

On the other hand, since the adjacent sides of a rhombus are of equal length, four isosceles triangles are determined:

DAB, BCD, CDA and ABC

Now the triangle (isosceles) theorem is invoked, which states that the angles adjacent to the base are of equal measure, from which it is concluded that:

δ ₁ = β2, δ2 = β ₁, α2 = γ ₁ and α ₁ = γ2 (**)

If the relations (*) and (**) are combined, the following equality of angles is reached:

α ₁ = α2 = γ ₁ = γ ₁ on the one hand and β ₁ = β2 = δ ₁ = δ2 on the other.

Recalling the equal triangles theorem that states that two triangles with an equal side between two equal angles are equal, we have:

AOD = AOB and consequently also the angles ∡AOD = ∡AOB.

Then ∡AOD + ∡AOB = 180º, but since both angles are of equal measure, we have 2 ∡AOD = 180º which implies that ∡AOD = 90º.

That is, it is shown geometrically that the diagonals of a rhombus intersect at right angles.

Exercises solved

- Exercise 1

Show that in a right trapezoid, the non-right angles are supplementary.

Solution

Figure 13. Right trapezoid. Source: F. Zapata.

The trapezoid ABCD is constructed with bases AB and DC parallel. The interior angle of vertex A is right (it measures 90º), so we have a right trapezoid.

The angles α and δ are internal angles between two parallels AB and DC, therefore they are equal, that is, δ = α = 90º.

On the other hand, it has been shown that the sum of the internal angles of a quadrilateral adds up to 360º, that is:

α + β + γ + δ = 90º + β + 90º + δ = 360º.

The above leads to:

β + δ = 180º

Confirming what was wanted to show, that the angles β and δ are supplementary.

- Exercise 2

A parallelogram ABCD has AB = 2 cm and AD = 1 cm, in addition the angle BAD is 30º. Determine the area of ​​this parallelogram and the length of its two diagonals.

Solution

The area of ​​a parallelogram is the product of the length of its base and its height. In this case, the length of the segment b = AB = 2 cm will be taken as the basis, the other side has length a = AD = 1 cm and the height h will be calculated as follows:

h = AD * Sen (30º) = 1 cm * (1/2) = ½ cm.

So: Area = b * h = 2 cm * ½ cm = 1 cm ².

References

1. CEA (2003). Geometry elements: with exercises and compass geometry. University of Medellin.
2. Campos, F., Cerecedo, FJ (2014). Mathematics 2. Grupo Editorial Patria.
3. Freed, K. (2007). Discover Polygons. Benchmark Education Company.
4. Hendrik, V. (2013). Generalized Polygons. Birkhäuser.
5. IGER. (sf). Mathematics First Semester Tacaná. IGER.
6. Jr. geometry. (2014). Polygons. Lulu Press, Inc.
7. Miller, Heeren, & Hornsby. (2006). Mathematics: Reasoning And Applications (Tenth Edition). Pearson Education.
8. Patiño, M. (2006). Mathematics 5. Editorial Progreso.
9. Wikipedia. Quadrilaterals. Recovered from: es.wikipedia.com