### 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:

### 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.

Solution:

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.