Angle Sum Property of Quadrilateral

We have already learned that three noncollinear points make a triangle on joining. Now let us mark four non-collinear points such that no three of them are collinear and see what we get by joining them. We get a closed figure with four sides by joining four non-collinear points, such a figure is called a quadrilateral.

All around us we see quadrilateral shapes like floor, ceiling, window, blackboard, table, and many more.

On this page, we will learn about quadrilaterals, their different properties, and the derivation of angle sum property of quadrilaterals.

Quadrilateral Definition

The word ‘ quad’ means four and the word ‘lateral’ means sides i.e. A quadrilateral is a figure bounded by four line segments such that no three of them are parallel.

A quadrilateral has four sides, four vertices, and four angles.

Thus below figure ABCD is a quadrilateral that is bounded by four sides i.e AB, BC, CD, and AD. The four vertices are A, B, C, and D. ∠A, ∠B, ∠C, and ∠D are the four angles of the quadrilateral. And it is written as □ABCD and read as quadrilateral ABCD.

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A line segment drawn from one vertex to the opposite vertex is called the diagonal of the quadrilateral. From the below figure Segment AC and BD are the diagonals of the quadrilateral ABCD

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Terms Related to Quadrilateral

Opposite Sides: Two sides of a quadrilateral are opposite sides if the sides have no common vertex. In the above figure side AB and DC; Side AD and BC are the two pairs of opposite sides.

Opposite Angles: Two angles of a quadrilateral are opposite angles if they don’t have any common arm. In the above figure, ∠A and ∠C; ∠B and ∠D, are two pairs of opposite angles.

Adjacent Sides: Two Sides of a quadrilateral are said to be adjacent if the sides have a common vertex. In the above figure Side AB and BC; Side BC and CD; Side CD and DA; Side DA and AB are the four pairs of adjacent sides also called consecutive sides.

Adjacent Angles: Two angles of a quadrilateral is said to be adjacent angles if the angles have a common side or an arm. In the above figure ∠A and ∠B; ∠B and ∠C, ∠C and ∠D; ∠D and ∠A are the four pairs of adjacent angles also called consecutive angles.

Types of Quadrilateral

There are basically six types of quadrilaterals. They are as follows,

1. Parallelogram: A quadrilateral that has its opposite sides congruent and parallel to each other is a parallelogram. Its opposite angles are also congruent to each other.

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2. Rectangle: A quadrilateral that has its opposite sides equal and all the angles are at right angles(900) is called a rectangle

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3. Square: A quadrilateral that has all its four sides equal and opposite side are parallel, and all the angles at right angles(900) is called a square.

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4. Rhombus: A quadrilateral has all its sides equal and its diagonals bisect each other at 900 is called a rhombus.

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5. Trapezium: A quadrilateral that has only one pair of sides are parallel and the two sides are non-parallel is called a trapezium. The sides may not be equal to each other.

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6. Kite: A quadrilateral is a kite if it has two pairs of equal adjacent sides and unequal opposite sides.

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Quadrilateral Angles

As we know that a quadrilateral has four angles. The sum of the angles of the quadrilateral is 3600.

The sum of all the angles of the □ABCD ∠A +∠B + ∠C + ∠D = 360°

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In the case of square and rectangle, the measure of all the angles is 900.

Therefore we have ∠A = ∠B = ∠C = ∠D = 90°

Angle Sum Property of Quadrilateral Theorem

Angle sum Property of a Quadrilateral Theorem states that,

The sum of the measures of four angles of a quadrilateral is 3600

that is, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.

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Angle sum property of quadrilateral Proof

Proof: Let ABCD be a quadrilateral and AC be the diagonal of the quadrilateral ABCD

From the figure, we can see that the diagonal AC divides the quadrilateral ABCD in two triangles i.e ABCand ADC

Step 1:

By the angle sum property, the sum of angles in a triangle is 180°.

So consider ABC,

∠B + ∠BCA + ∠BAC= 180° ……(1)

Similarly in ADC

∠D + ∠DCA + ∠DAC = 180° …..(2)

Step 2:

Adding equation 1 and 2 we get

(∠B + ∠BCA + ∠BAC) + (∠D + ∠DCA + ∠DAC) = 180 + 180

(∠B + ∠BCA + ∠BAC) + (∠D + ∠DCA + ∠DAC) = 3600………..(3)

Step 3:

From the figure,

∠DAC + ∠BAC = ∠DAB

∠BCA + ∠DCA = ∠BCD.

Step 4:

Putting these values in equation 3

∠D + ∠BCD + ∠DAB + ∠B = 360°

That is, ∠D + ∠C + ∠A + ∠B = 360°.

∠A + ∠B + ∠C + ∠D = 360°.

Hence the angle sum property of the quadrilateral theorem proved.

The sum of the measures of four angles of a quadrilateral is 3600 Using the angle sum property of the quadrilateral theorem let us solve some examples.

Solved Examples

Example 1:

In the figure below x = 800, y = 1100, z =1200. Find w?

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Solution:

From Angle sum property of quadrilateral Proof

We have w + x + y + z = 3600

x = 800, y = 1100, z =1200

w + 80 + 110 + 120 = 360

w + 310 = 360

w = 360 - 310

w = 500

Example 2: Find the value of x

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Solution:

From Angle sum property of quadrilateral Proof

We have, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.

Given ∠ABC =1060, ∠BCD = x0, ∠CDA = 680, ∠DAB = 1260

So, 106 + x + 68 + 126 = 360

x + 300 = 360

x = 360 - 300

x = 600

Quiz

Using angle sum property of quadrilateral theorem solve the following

The three angles of a quadrilateral are 60˚, 70˚, 90˚. Find the fourth angle?

If the measure of two angles of a quadrilateral is 65˚ and 85˚ and the other two angles are equal, find the measure of each of the equal angles?

FAQ (Frequently Asked Questions)

1. What is a Cyclic Quadrilateral?

Answer: A quadrilateral that is formed by joining the four points on a circle is called a cyclic quadrilateral. Or you can say that a quadrilateral is said to be a cyclic quadrilateral if there exists a circle passing through all its four vertices. The sum of the measures on the four angles of a quadrilateral is 3600. The opposite angles are supplementary.

From the below figure quadrilateral ABCD is a cyclic quadrilateral.

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2. What is the Area of a Quadrilateral?

Answer: A quadrilateral is a closed figure with four sides. According to the sides of the quadrilateral area, the different quadrilateral is calculated by different formulas.

Area of a Square | Square Area = (side) |

Area of a Kite | Kite Area = ( ½ ) × Product of Diagonals |

Area of a Parallelogram | Parallelogram Area = Base × Height |

Area of a Rectangle | Rectangle Area = Length × Breadth |

Area of a Trapezoid | Trapezoid Area = (base |