Angle Sum of a Quadrilateral

Just as triangles have an angle-sum of $180^\circ$, quadrilaterals have an angle-sum of $360^\circ$. The proof is one line longer than the triangle proof: draw a diagonal, count the angles in each of the two triangles formed, and add. This lesson establishes the result and surveys the named families of quadrilaterals we will study.

Definitions

A quadrilateral is a closed plane figure bounded by four line segments , the sides , meeting only at their endpoints , the vertices. The segments connecting opposite vertices are the diagonals; every quadrilateral has two diagonals.

A quadrilateral is convex if both diagonals lie inside it. (We work mainly with convex quadrilaterals in this chapter.)

The theorem and its proof

Theorem (Angle sum of a quadrilateral). The sum of the four interior angles of a convex quadrilateral is $360^\circ$.

Proof. Let $ABCD$ be a convex quadrilateral. Draw the diagonal $AC$, which splits $ABCD$ into two triangles: $\triangle ABC$ and $\triangle ACD$.

By the triangle angle sum theorem: $\angle BAC + \angle ABC + \angle BCA = 180^\circ,$ $\angle CAD + \angle ACD + \angle ADC = 180^\circ.$

Add the two equations: $\angle BAC + \angle CAD + \angle ABC + \angle BCA + \angle ACD + \angle ADC = 360^\circ.$

Group adjacent angles at $A$ and $C$: $(\angle BAC + \angle CAD) + \angle ABC + (\angle BCA + \angle ACD) + \angle ADC = \angle A + \angle B + \angle C + \angle D = 360^\circ.$

Q.E.D.

So in any convex quadrilateral, $\angle A + \angle B + \angle C + \angle D = 360^\circ$.

Named quadrilaterals , a brief tour

Quadrilaterals come in a family tree of named special cases. Here is the gallery:

• Trapezium (Indian definition): a quadrilateral with at least one pair of parallel sides. (Note: in some countries this figure is called a "trapezoid".)
• Parallelogram: a quadrilateral with both pairs of opposite sides parallel. Every parallelogram is automatically a trapezium.
• Rectangle: a parallelogram with one right angle (hence all four are right angles).
• Rhombus: a parallelogram with two adjacent sides equal (hence all four sides equal).
• Square: a quadrilateral that is both a rectangle and a rhombus. All angles right and all sides equal.
• Kite: a quadrilateral with two distinct pairs of consecutive sides equal.

The family tree: $\text{Square} \subset \text{Rectangle} \subset \text{Parallelogram} \subset \text{Trapezium} \subset \text{Quadrilateral}.$ Also $\text{Square} \subset \text{Rhombus} \subset \text{Parallelogram}$.

So every square is a rectangle (with extra constraints) and every square is a rhombus.

Using the angle sum

The simplest application: given three of the four angles of a quadrilateral, find the fourth.

Example. If $\angle A = 80^\circ, \angle B = 100^\circ, \angle C = 110^\circ$ in quadrilateral $ABCD$, then $\angle D = 360 - 80 - 100 - 110 = 70^\circ$.

Slightly fancier: angles in ratio.

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

Let them be $k, 2k, 3k, 4k$. Sum: $10k = 360 \Rightarrow k = 36$. Angles: $36^\circ, 72^\circ, 108^\circ, 144^\circ$.

Worked examples

Example 1. Three angles of a quadrilateral are $100^\circ, 75^\circ, 60^\circ$. Find the fourth.

$\angle 4 = 360 - 100 - 75 - 60 = 125^\circ$.

Example 2. Two angles of a quadrilateral are equal, and the remaining two are $80^\circ$ and $120^\circ$. Find the equal angles.

$\angle 1 + \angle 2 = 360 - 80 - 120 = 160$. Each: $80^\circ$.

Example 3. The angles of a quadrilateral are in ratio $2 : 3 : 4 : 6$. Find them.

Sum of ratios $= 15$. Each "part" $= 360/15 = 24$. Angles: $48, 72, 96, 144$.

Example 4. A convex quadrilateral has three right angles. Find the fourth.

$\angle 4 = 360 - 3 \cdot 90 = 90^\circ$. All four are right angles , the quadrilateral is a rectangle.

Example 5. Each angle of a quadrilateral is equal. Find the value.

$4x = 360 \Rightarrow x = 90^\circ$. Each angle is a right angle , the quadrilateral is a rectangle (or square).

Try it yourself

1. State the angle sum of a quadrilateral and prove it.
2. Three angles of a quadrilateral are $50^\circ, 70^\circ, 100^\circ$. Find the fourth.
3. The angles are in ratio $1 : 1 : 2 : 2$. Find them.
4. Can a quadrilateral have four obtuse angles?
5. Define rectangle, rhombus, square, kite, and trapezium.
6. Is every rectangle a parallelogram?
7. Is every square a rhombus?
8. The angles of a quadrilateral are $(3x + 10)^\circ, (5x - 20)^\circ, (2x + 30)^\circ, (4x)^\circ$. Find $x$.
9. Construct an example of a quadrilateral with exactly one pair of parallel sides.
10. Why is a square a special case of both a rectangle and a rhombus?

Pitfalls / Insight

• All four angles count , interior, not exterior. Be careful when reading figures.
• The figure must be convex. Concave quadrilaterals technically still have angle sums of $360^\circ$ when measured correctly, but some students get confused.
• Different textbooks have different definitions of "trapezium". In India, a trapezium has at least one pair of parallel sides (so parallelograms count). In some other countries, exactly one pair (so parallelograms don't count). We use the Indian definition.

Insight. The $360^\circ$ angle sum is the gateway to all the parallelogram theorems. Combined with the parallel-line theorems from Chapter 6, every claim about a parallelogram's angles can be proved in a few lines.