Angle Sum Property of a Quadrilateral: Definition, Properties, Proofs

A quadrilateral is a polygon with four sides, four interior angles and eight exterior angles. There are various types of quadrilaterals and all of them follow the angle sum property of quadrilaterals. The angle sum property of a quadrilateral states that the sum of all interior angles of a quadrilateral is \(360^\circ \). In this article we have provided a detailed definition of this property with proof. Moreover, we discuss the sum property of a polygon and triangle as well. Read on to learn more about the Angle Sum Property of a Quadrilateral.

Angle Sum Property of a Quadrilateral

Before explaining what the angle sum property of a quadrilateral is, let us first understand what quadrilaterals are. We encounter quadrilaterals everywhere in life. Any shape with four sides including all squares and rectangles are quadrilaterals. When four non-collinear points take up a shape, it is called a quadrilateral. A quadrilateral has four sides, four angles, and four vertices.

Both the figures given above are quadrilaterals. ABCD is a quadrilateral. AB, BC, CD, and DA are the four sides of the quadrilateral. A, B, C, and D are the four vertices, and ∠A, ∠B, ∠C, and ∠D are the angles of the quadrilateral.

Angle Sum Property of a Quadrilateral states that the sum of all angles of a quadrilateral is 360°

Angle Sum Property of Polygon

A polygon is a simple closed two-dimensional shape formed by joining the straight line segments. Examples of polygons are triangle, quadrilateral, pentagon, hexagon, etc. The important points related to the angles of a polygon are:

1. The formula for calculating the sum of interior angles is \(\left({n – 2} \right) \times 180^\circ \) or \(\left({2n – 4} \right) \times 90^\circ \) where n is the number of sides.

2. All the interior angles of a regular polygon are equal. The formula for calculating the measure of an interior angle of a polygon is given by:

\({\text{Interior}}\,{\text{angle}}\,{\text{of}}\,{\text{a}}\,{\text{polygon}} = \frac{{{\text{ Sum of interior angles }}}}{{{\text{ Number of sides }}}}\)

3. The sum of all the exterior angles of a polygon is \(360^\circ \).

4. The formula for calculating the measure of an exterior angle is given by

\({\text{Exterior}}\,{\text{angle}}\,{\text{of}}\,{\text{a}}\,{\text{polygon}} = \frac{{360^\circ }}{{{\text{ Number of sides }}}}\)

Angle Sum Property of Triangle

A triangle is the smallest polygon formed by three line segments, making the interior and exterior angles. An interior angle is an angle formed between two adjacent sides of a triangle. In contrast, an exterior angle is an angle formed between a side of the triangle and an adjacent side extending outward.

We know that a triangle is a polygon with three sides, so, \(n=3\).
Thus, using the formula of calculating the sum of interior angles, we get the sum of interior angles of a triangle asInterior angle sum \(\; = \left( {3 – 2} \right) \times 180^\circ \; = 180^\circ \)

There are different types of triangles, but for each type, the sum of the interior angles is \(180^\circ \). According to the angle sum property of a triangle, the sum of all three interior angles of a triangle is \(180^\circ \). The angle sum property of a triangle is useful for finding the measure of an unknown angle when the values of the other two angles are known.

Proof for the Sum of the Interior Angles of a Triangle is \(180^\circ \)

To prove: Sum of the interior angles of a triangle is \(180^\circ \)
Let us consider a \(\Delta ABC\).

Construction:

 In \(\Delta ABC\) given above, a line is drawn parallel to the side \(BC\) of \(\Delta ABC.\)

This line passes through vertex \(A\). Label this line as \(PQ\).

Since the straight angle measures \(180^\circ \),
\(\angle PAQ = 180^\circ \)

That is,

\(\angle PAB + \angle BAC + \angle CAQ = 180^\circ ….\left( 1 \right)\)

As \(PQ\|BC,\,AB\) is a transversal, and the alternate interior angles are equal.
\(\therefore \angle PAB = \angle ABC…\left(2\right)\)

Similarly, as \(PQ||BC\) and \(AC\) is a transversal,

\(\angle CAQ = \angle ACB\quad \ldots ..(3)\)

Now, using equations \(2\) and \(3\) marked above, substitute \(\angle ABC\) for \(\angle PAB\) and \(\angle ACB\) for \(\angle CAQ\) in equation \(1\): 

\(\angle ABC + \angle BAC + \angle ACB = 180^\circ \ldots ..(4)\)

Hence, if we consider \(\Delta ABC\), equation \((4)\) implies that the sum of the interior angles of \(\Delta ABC\) is \(180^\circ \). We can also write this as

\(\angle A+\angle B+\angle C=180^{\circ} .\).

Thus, it is proved that the sum of all the interior angles of a triangle is \(180^\circ \).

Quadrilateral

The word quadrilateral is derived from the two Latin words: ‘quadri’ means four and ‘latus’ means sides. A quadrilateral is a two-dimensional shape having four sides, four angles, and four corners or vertices. The sum of internal angles of a quadrilateral is \(360^\circ \).

Angle Sum Property of a Quadrilateral

A quadrilateral is a \(4-\) sided polygon made up of all line segments.

So, \(n=4\)

Thus, using the formula of angle sum property of a polygon, we get

Interior angle sum \(=(4-2) \times 180^{\circ}=2 \times 180^{\circ}=360^{\circ}\)

Proof of Angle Sum Property of a Quadrilateral

We can use the angle sum property of the triangle to find the sum of the interior angles of another polygon. Since every polygon can be divided into triangles, the angle sum property can be extended to find the sum of the angles of all polygons. Let us see how this is applicable in quadrilaterals.

Let us prove that the sum of all the four angles of a quadrilateral is \(360^\circ \).

To prove: \(\angle ADC + \angle DAB + \angle BCD + \angle ABC = 360^\circ \)
Construction: Join \(A\) and \(C\)
Given, \(\angle ADC,\angle DAB,\angle BCD,\angle ABC\) are four interior angles of quadrilateral \(ABCD\) and \(AC\) is the diagonal constructed.

We know that the sum of angles in a triangle is \(180^\circ \).

Now, consider \(\Delta ADC\),

\(\angle ADC + \angle DAC + \angle DCA = 180^\circ \ldots \ldots (1)\)  (Sum of the interior angles of a triangle)

Now, consider triangle \(\Delta ABC\),

\(\angle ABC + \angle BAC + \angle BCA = 180^\circ \ldots .(2)\)  (Sum of the interior angles of a triangle)

On adding both equations \((1)\) and \((2)\), we have,

\((\angle ADC + \angle DAC + \angle DCA) + (\angle ABC + \angle BAC + \angle BCA) = 180^\circ + 180^\circ \)

\(\Rightarrow \angle ADC + (\angle DAC + \angle BAC) + (\angle BCA + \angle DCA) + \angle ABC = 360^\circ \ldots (3)\)

We see that \((\angle DAC + \angle BAC) = \angle DAB\) and \((\angle BCA + \angle DCA) = \angle BCD\).

Substituting them in equation \((3)\) we have,

\(\angle A D C+\angle D A B+\angle B C D+\angle A B C=360^{\circ}\)

Hence, it proved the angle sum property of the quadrilateral.

Solved Examples – Angle Sum Property of a Quadrilateral

Q.1. If the sum of three interior angles of a quadrilateral is \(240^\circ \), find the fourth angle.
Ans: Given that the sum of three interior angles of a quadrilateral is \(240^\circ \).
Let us assume the fourth angle as \(x\).
We know that sum of four interior angles of a quadrilateral is \(360^\circ \).
Thus, \(x + 240^\circ = 360^\circ \)
\( \Rightarrow x = 360^\circ – 240^\circ = 120^\circ \)
Hence, the fourth angle is \(120^{\circ}\).

Q.2. If the angles of a quadrilateral are in the ratio \(6:3:4:5\), determine the value of the four angles.
Ans: Let the angles be \(6x, 3x, 4x\), and \(5x\).
According to the angle sum property of the quadrilateral,
\(6x + 3x + 4x + 5x = 360^\circ \)
\(\Rightarrow 18 x=360^{\circ}\)
\( \Rightarrow x = 20^\circ \)
Thus, the four angles will be, \(6x = 6 \times 20^\circ = 120^\circ \)
\(3x = 3 \times 20^\circ = 60^\circ ,4x = 4 \times 20^\circ = 80^\circ ,5x = 5 \times 20^\circ = 100^\circ \)
Therefore, the four angles are \(120^\circ ,60^\circ ,80^\circ ,100^\circ \).

Q.3. In a quadrilateral, if the sum of two angles is 200°, find the measure of the other two equal angles.
Ans: Given, the sum of two angles is \(200^\circ \).
Let us say the measure of equal angles is \(x\).
We know the sum of the interior angles of a quadrilateral is \(360^\circ \).
We can say, \(x + x + 200^\circ = 360^\circ \Rightarrow 2x = 360^\circ – 200^\circ \Rightarrow x = \frac{{160^\circ }}{2} = 80^\circ \)
Therefore, the measure of equal angles is \(80^\circ \).

Q.4. If three angles of a quadrilateral are equal and the measure of the fourth angle is \(30^\circ \), find the measure of each of the equal angles?
Ans: Let the measure of each of the equal angles be \(x\).
According to the angle sum property of a quadrilateral, the sum of all angles of a quadrilateral \( = 360^\circ \)
\(30^\circ + x + x + x = 360^\circ \)
\( \Rightarrow 30^\circ + 3x = 360^\circ \Rightarrow 3x = 360^\circ – 30^\circ \Rightarrow 3x = 330^\circ \)
\(\Rightarrow x = \frac{{330^\circ }}{3}\)
\( \Rightarrow x = 110^\circ \)
Hence, the measure of each equal angle is \(\Rightarrow x=110^{\circ}\).

Q.5. If one angle of a quadrilateral is double of another angle and the measure of the other two angles are \(60^\circ,\,80^\circ \). Find the measurement of the unknown angles.
Ans: According to the angle sum property of a quadrilateral,
The sum of all angles of a quadrilateral \( = 360^\circ \)
Let us say one unknown angle is \(x\) and the other unknown angle is \(2x\).
\(60^\circ + 80^\circ + x + 2x = 360^\circ \)
\(\Rightarrow 140^\circ + 3x = 360^\circ \Rightarrow 3x = 360^\circ – 140^\circ \Rightarrow 3x = 120^\circ \)
\(\Rightarrow x = \frac{{120^\circ }}{3} = 40^\circ \)
\( \Rightarrow x = 40^\circ ,\,2x = 40^\circ  \times 2 = 80^\circ \)
Therefore, the unknown angles are \(40^\circ ,\,80^\circ \).

Study About Angle Sum Property of Triangle

FAQs on Angle Sum Property of a Quadrilateral

The answers to some of the most frequently asked questions on Angle Sum Property of a Quadrilateral are given below: