What is a quadrilateral?

A quadrilateral is the simplest closed shape after the triangle. It has four sides and four corners , quad meaning "four" and latus meaning "side". The word has been in mathematical use for over $2000$ years, and the shape is just as old.

Concept

Formally, a quadrilateral $ABCD$ is the closed figure formed by joining four points $A, B, C, D$ (no three of which are collinear) in order, edge-to-edge. The four line segments $AB, BC, CD, DA$ are the sides. The four points $A, B, C, D$ are the vertices. The angles $\angle A, \angle B, \angle C, \angle D$ inside the figure are the interior angles. The two segments $AC$ and $BD$ connecting opposite vertices are the diagonals.

Notation. When naming a quadrilateral, the order of letters matters: we go around the shape. So "$ABCD$" means side $AB$ joins to side $BC$ joins to $CD$ joins to $DA$. The diagonals are then automatically $AC$ and $BD$.

Convex vs concave. A quadrilateral is convex if both diagonals lie inside the figure (and equivalently, each interior angle is less than $180^\circ$). It is concave if one of the diagonals goes outside , one of the angles is then a reflex angle, greater than $180^\circ$. The dart shape is a familiar concave quadrilateral.

The angle sum. The sum of the four interior angles of any quadrilateral , convex or concave , is exactly $360^\circ$.

Why? Draw a diagonal, say $AC$. It splits the quadrilateral into two triangles, $\triangle ABC$ and $\triangle ACD$. Each triangle's angle sum is $180^\circ$, and together their six angles add up to the four interior angles of the quadrilateral. So:

$\angle A + \angle B + \angle C + \angle D = 180^\circ + 180^\circ = 360^\circ.$

This single fact lets us compute the missing angle in any quadrilateral if we know the other three.

Sides and diagonals. A quadrilateral has $4$ sides and $\dbinom{4}{2} - 4 = 2$ diagonals. As the number of sides grows the number of diagonals grows much faster: a pentagon has $5$, a hexagon has $9$, an $n$-gon has $\dfrac{n(n-3)}{2}$.

Adjacent and opposite. In quadrilateral $ABCD$:

• Sides $AB$ and $BC$ are adjacent (share vertex $B$). Sides $AB$ and $CD$ are opposite.
• Angles $\angle A$ and $\angle B$ are adjacent. Angles $\angle A$ and $\angle C$ are opposite.

These words appear in every property of the special quadrilaterals.

Worked examples

Example 1. Three angles of a quadrilateral are $80^\circ, 100^\circ$ and $110^\circ$. Find the fourth.

• Sum $= 360^\circ$.
• $80 + 100 + 110 = 290$.
• Fourth angle $= 360 - 290 = 70^\circ$.

Example 2. The four angles of a quadrilateral are in the ratio $1 : 2 : 3 : 4$. Find each.

• Let them be $x, 2x, 3x, 4x$.
• Sum: $x + 2x + 3x + 4x = 10x = 360^\circ$.
• $x = 36^\circ$. So the angles are $36^\circ, 72^\circ, 108^\circ, 144^\circ$.

Example 3. In quadrilateral $PQRS$, $\angle P = 90^\circ$, $\angle Q = 60^\circ$, $\angle R = 120^\circ$. Find $\angle S$.

• $\angle S = 360^\circ - (90+60+120) = 360 - 270 = 90^\circ$.

Example 4. Decide whether the angles $100^\circ, 80^\circ, 100^\circ, 100^\circ$ can be the four angles of a quadrilateral.

• Sum $= 100+80+100+100 = 380^\circ \ne 360^\circ$.
• So no, these cannot be the four angles.

Try it yourself

1. Three angles of a quadrilateral are $75^\circ, 90^\circ$ and $115^\circ$. Find the fourth.
2. The four angles are in the ratio $3:4:5:6$. Find them.
3. Can a quadrilateral have all four angles obtuse? Justify.
4. Draw a concave quadrilateral and label its reflex angle.
5. In quadrilateral $WXYZ$, $\angle W = \angle Y$ and $\angle X = \angle Z$. If $\angle W = 70^\circ$, find $\angle X$.
6. A quadrilateral has angles $a, a, 110^\circ, 130^\circ$. Find $a$.
7. List all pairs of (i) adjacent sides and (ii) opposite sides in quadrilateral $KLMN$.
8. Can the diagonals of every quadrilateral meet inside? Justify with an example.

Activity

Snip and rearrange. Cut out any quadrilateral from coloured paper. Tear off the four corners (without losing them!) and place them edge-to-edge on a flat surface. They should fit together to form a full turn around a point , that is exactly $360^\circ$, the angle sum of a quadrilateral. This is the same hands-on demonstration used for the triangle's $180^\circ$ angle sum, scaled up.