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Quadrilateral: Angle Sum Property

Last updated date: 17th Apr 2024
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Prove that the Sum of the Angles of a Quadrilateral is 360 Degree

Prior to discussing the quadrilaterals angle sum property, let us review what angles and quadrilaterals are. The angle is shaped when a two-line fragment joins at a solitary point. An angle is evaluated in degrees (°). Quadrilateral angles are the angles plot inside the state of a quadrilateral. The quadrilateral is a four-sided polygon which can have or not have practically identical sides. It is a shut figure in two-assessment and has non-bent sides.

What is a Quadrilateral

A quadrilateral is a polygon which has 4 vertices and 4 sides encasing 4 angles and the sum of the huge number of angles is 360°. Precisely when we draw the diagonals to the quadrilateral, it structures two triangles. Both these triangles bear an angle sum of 180°. Subsequently, the incomparable angle sum of the quadrilateral is 360°. Angle sum constitutes one of the properties of quadrilaterals. In this article, we will get capacity with the guidelines of angle sum property.

Sorts of Quadrilaterals

There are generally five sorts of quadrilaterals. They are;

  • Parallelogram: 

Which has converse sides as the same and comparing to each other.

  • Rectangle: 

Which has comparable reverse sides anyway all the angles are at 90 degrees.

  • Square: 

Which all its four sides are the same and angles at 90 degrees.

  • Rhombus: 

Its a parallelogram with all of its sides the same and its diagonals isolate each other at 90 degrees.

  • Trapezium: 

Which has only one arrangement of sides as equivalent and the sides may not be identical to one another.

Angle Sum Property of a Quadrilateral

As shown by the angle sum property of a Quadrilateral, the sum of the huge number of four inside angles is 360 degrees.

Confirmation: In the quadrilateral named PQRS,

∠PQR, ∠QRS, ∠RSP, and ∠SPQ are the internal angles.

QR is a diagonal

QR disengages the quadrilateral into two triangles, ∆PQR and ∆PSR

We have found that the sum of internal angles of a quadrilateral is 360°, that is, ∠PQR + ∠QRS + ∠RSP + ∠SPQ = 360°.

We should exhibit that the sum of the overall huge number of four angles of a  Quadrilateral is 360 degrees.

We understand that the sum of angles in a triangle is 180°.

As of now consider triangle PSR,

∠S + ∠SPR + ∠SRP = 180° (Sum of angles in a triangle)

As of now consider triangle PQR,

∠Q + ∠QPR + ∠QRP = 180° (Sum of angles in a triangle)

On adding both the conditions procured above we have,

(∠S + ∠SPR + ∠SRP) + (∠Q + ∠QPR + ∠QRP) = 180° + 180° 

∠S + (∠SPR + ∠QPR) + (∠QRP + ∠SRP) + ∠Q = 360°

We see that (∠SPR + ∠QPR) = ∠SPQ and (∠QRP + ∠SRP) = ∠QRS.

Displacing them we have,

∠S + ∠SPQ + ∠QRS + ∠Q = 360° 

That is,

∠S + ∠P + ∠R + ∠Q = 360°.

Or of course, the sum of angles of a quadrilateral is 360°. This is the angle sum property of quadrilaterals.

What is a Quadrilateral Equation

Area of the quadrilateral is the absolute space involved by the figure. The zone recipe for the various quadrilaterals is given underneath:

Area of a Parallelogram

Base x Height

Area of a Rectangle

Length x Width

Area of a Square

Side x Side

Area of a Rhombus

(1/2) x Diagonal 1 x Diagonal 2

Area of a Kite

1/2 x Diagonal 1 x Diagonal 2

Solved Example

Example: The angles of a quadrilateral are as follows

(3a + 2)°

(a – 3)

(2a + 1)°

2(2a + 5) °

Calculate the value of ‘a’ and how much each angle measures.


Applying the angle sum property of quadrilateral, we obtain

(3a + 2)°+ (a – 3)° + (2a + 1)° +  2(2a + 5)°= 360°                 

⇒ 3a + 2 + a - 3 + 2a + 1 + 4a + 10 = 360°

⇒ 10a + 10 = 360                              

⇒ 10a = 360 – 10                              

⇒ 10a = 350                       

⇒ a = 350/10                     

⇒ a = 35                                              

Thus, (3a + 2)

= 3 × 35 + 2

= 105 + 2 = 107°

(a – 3) = 35 – 3 = 32°

(2a + 1) = 2 × 35 + 1 = 70 + 1 = 71°

2(2a + 5)

= 2(2 × 35 + 5)

= 2(70 + 5)

= 2 × 75 = 150°

Hence, the four angles of the quadrilateral measures 32°, 150°,107°, 71°  respectively.

FAQs on Quadrilateral: Angle Sum Property

Q1: What is an Angled and Internal Quadrilateral?

Answer: A raised quadrilateral can be portrayed as a quadrilateral whose both the diagonals are completely contained inside the figure. On the other hand, a bended quadrilateral is a quadrilateral which is having in any occasion one awry that lies deficiently or through and through outside of the figure.

Q2: What is the Sum of within Angles of a Quadrilateral?

Answer: The sum of the general huge number of inside angles of a quadrilateral is 360°.

Q3: What are the Three Credits of a Quadrilateral?

Answer: The three critical qualities of a quadrilateral are:

  • Four sides

  • Four Vertices

  • the sum of within angles should be identical to 360 degrees.

Q4: How to Find the Edge of a Quadrilateral?

Answer: The edge of a quadrilateral can be constrained by adding the side length of the overall huge number of four sides.