Quadrilaterals

What is a Quadrilateral?

The abbreviation "quad" describes something that is in terms of four. For example, a mother carrying four babies in one pregnancy will be expecting the birth of quadruplets. In geometry, this contraction is found in the term quadrilateral, which adopts the same idea. Let's define it below.

Naming a Quadrilateral

It is important to note that the order of the vertices for a given quadrilateral must be taken into account when naming a quadrilateral. Here is an example.

Types of Quadrilaterals

There are six types of quadrilaterals we should familiarise ourselves with throughout this topic. This is described in the table below.

Properties of Quadrilaterals

Looking at our previous table, there are four characteristics all quadrilaterals have in common. Listing them below, we can say that every quadrilateral has

1. Four sides;

2. Four vertices;

3. Two diagonals;

4. The sum of all their interior angles is 360^o.

Although all quadrilaterals have the same basic qualities, the measures of their sides and angles differ from one another. In the table below, we shall describe the distinct features of the six quadrilaterals introduced before.

Angles of Quadrilaterals

In this segment, we shall discuss the characteristics of the interior and exterior angles of a quadrilateral. We shall also observe some notable relationships between their opposite and adjacent angles.

Interior Angles of a Quadrilateral

We had mentioned before that the sum of the interior angles of a quadrilateral is $360º$. Since a quadrilateral is a type of polygon, we can prove this statement by using the following formula.

Sum of Interior Angles Formula

For a polygon with n number of sides, the sum of its interior angles, S is equal to

$$S=(n-2)\times 180º$$

Since a quadrilateral has four sides, then $n=4$ and so

$$S=(4-2)\times 180º$$

$$⟹S=2\times 180º$$

$$⟹S=360º$$

Thus, we have shown that all the interior angles of a quadrilateral add up to $360º$.

Interior angles of a quadrilateral, StudySmarter Originals

In the rectangle above, $\mathrm{\angle }SPQ+\mathrm{\angle }PQR+\mathrm{\angle }QRS+\mathrm{\angle }RSP=360º$.

Adjacent Angles of Quadrilaterals

Two angles that lie side by side are called adjacent angles. Adjacent angles often make use of two important concepts, namely

1. Supplementary angles: Two adjacent angles are supplementary when the sum of their measures is equal to $180º$.

2. Complementary angles: Two adjacent angles are complementary when the sum of their measures is equal to $90º$.

Both these ideas can be applied to quadrilaterals as well. Let us observe the rhombus below.

Supplementary angles of a quadrilateral, StudySmarter Originals

The straight line segment $CBX$ is made up of $\mathrm{\angle }ABC$ (interior angle of the rhombus) and $\mathrm{\angle }ABX$ (exterior angle of the rhombus). The angle of a straight line is always $180º$. Since $\mathrm{\angle }ABC$ and $\mathrm{\angle }ABX$ lie on this line and are adjacent to each other, they are supplementary angles. Say we are given $\mathrm{\angle }CBA=114º$. To find $\mathrm{\angle }ABX$, we can conduct the following calculation.

$$\mathrm{\angle }ABC+\mathrm{\angle }ABX=\overline{CBX}$$

$$⟹\mathrm{\angle }ABC+114º=180º$$

$$⟹\mathrm{\angle }ABC=180º-114º$$

$$⟹\mathrm{\angle }ABC=66º$$

Thus, $\mathrm{\angle }ABX=66º$. Now, let us take a look at the rectangle below.

Complementary angles of a quadrilateral, StudySmarter Originals

Given the properties of a rectangle, we know that each interior angle is $90º$. Say $\mathrm{\angle }RSQ=22º$. We are now told to find $\mathrm{\angle }PSQ$. To do this, simply note that the sum of $\mathrm{\angle }RSQ$ and $\mathrm{\angle }PSQ$ equal $90º$ since they are adjacent and lie at a right angle. Thus, $\mathrm{\angle }PSQ$ is $68º$ since

$$\mathrm{\angle }PSQ+\mathrm{\angle }RSQ=\mathrm{\angle }PSR$$

$$⟹\mathrm{\angle }PSQ+22º=180º$$

$$⟹\mathrm{\angle }PSQ=180º-22º$$

$$⟹\mathrm{\angle }PSQ=68º$$

Exterior Angles of a Quadrilateral

The sum of the exterior angles of any polygon equals $360º$. This means that the sum of the exterior angles of a quadrilateral is also $360º$. To show this, let us return to our previous rectangle and draw an extended straight line at each vertex.

Exterior angles of a quadrilateral, StudySmarter Originals

The interior angles of this rectangle are $\mathrm{\angle }SPQ$, $anglePQR$, $\mathrm{\angle }QRS$ and $\mathrm{\angle }RSP$, while the exterior angles are $\mathrm{\angle }W$, $\mathrm{\angle }X$, $\mathrm{\angle }Y$ and $\mathrm{\angle }Z$. Notice that each interior angle corresponds to an exterior angle that is adjacent to them and lies on a straight line. Recall that the angle of a straight line is $180º$. Thus, we have four sets of supplementary angles. The exterior angles in this case are also equal since the interior angles are all equal. Let's solve one of these exterior angles to prove our claim.

$$\mathrm{\angle }PQR=\mathrm{\angle }X=\overline{RQA}$$

$$⟹\mathrm{\angle }PQR+90º=180º$$

$$⟹\mathrm{\angle }PQR=180º-90º$$

$$⟹\mathrm{\angle }PQR=90º$$

Since, $\mathrm{\angle }X=\mathrm{\angle }Y=\mathrm{\angle }Z=\mathrm{\angle }W=90º$, then $\mathrm{\angle }X+\mathrm{\angle }Y+\mathrm{\angle }Z+\mathrm{\angle }W=9(90º)=360º$, as required.

Vertically Opposite Angles of Quadrilaterals

Vertically opposite angles are two (opposite) angles constructed by two straight intersecting lines that extend from a pair of distinct vertices. Vertically opposite angles are always equal to each other. Refer back to our rectangle above with O being the point of intersection between the two diagonals.

Vertically opposite angles of a quadrilateral, StudySmarter Originals

Here, $\mathrm{\angle }POQ$ and $\mathrm{\angle }SOR$ are vertically opposite angles. Thus, $\mathrm{\angle }POQ=\mathrm{\angle }SOR$. Similarly, $\mathrm{\angle }POS$ and $\mathrm{\angle }QOR$ are vertically opposite angles and so $\mathrm{\angle }POS=\mathrm{\angle }QOR$. The same concept applies to squares, rectangles, parallelograms, rhombuses and kites (try it yourself!).

Corresponding Angles of Quadrilaterals

Corresponding angles are formed when two parallel lines are intersected by the transversal line. A transversal line is a line that crosses a pair of lines on the same plane at two specific points. Corresponding angles are always equal to each other. Let's show this with the parallelogram below.

Corresponding angles of a quadrilateral, StudySmarter Originals

Here, $\mathrm{\angle }ADC$ (interior angle of the parallelogram) and $\mathrm{\angle }XAB$ (exterior angle of the parallelogram) are corresponding angles since the line $DAX$ passes through parallel sides $AB$ and $DC$. Thus, $\mathrm{\angle }ADC=\mathrm{\angle }XAB$.

Similarly, $\mathrm{\angle }BCD$ and $\mathrm{\angle }YBA$ are corresponding angles since the line $CBY$ crosses the parallel sides $AD$ and $BC$. Therefore, $\mathrm{\angle }BCD=\mathrm{\angle }YBA$. This concept applies to squares, rectangles and rhombuses as well (give it a go!).

Perimeter of Quadrilaterals

The perimeter of a quadrilateral is defined as the total length of its boundary. In other words, it is the sum of all its sides. Therefore, for a quadrilateral $ABCD$

The perimeter of quadrilaterals, StudySmarter Originals

with sides $AB$, $BC$, $CD$ and $DA$, the perimeter, $P$ is

$$P=\overline{AB}+\overline{BC}+\overline{CD}+\overline{DA}$$

or

$$P=a+b+c+d$$

Here are two examples.

Area of Quadrilaterals

The area of a quadrilateral is described by the space enclosed within its boundary. Each of the six types of quadrilaterals we have seen above has its own area formula. The table below presents their respective area formulas.

Let us look at two worked examples.

Examples Involving Quadrilaterals

We shall end this topic with several worked examples of quadrilaterals that make use of the concepts we have established throughout this discussion.