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In this article we are going to learn about the Sum of the Four Sides of a Quadrilateral Exceeds the Sum of the Diagonals. We have to prove that the sum of all the four sides of a quadrilateral is greater than the sum of the diagonals.

## Sum of the Four Sides of a Quadrilateral Exceeds the Sum of the Diagonals Theorem

**Theorem Statement:**

Prove that the sum of the sides of a quadrilateral is greater than twice the length of its diagonal?

**Proof:**

Given PQRS is a quadrilateral, PR and QS are its diagonals

To Prove:

(PQ + QR + RS + SP) > ( PR + QS)

Proof:

In ∆PSQ

(SP + PQ) > QS [ therefore the sum of the two sides of a triangle is greater than the third side]

In ∆PQR

(PQ + QR) > PR [ therefore the sum of the two sides of a triangle is greater than the third side]

In ∆QRS

(QR + RS) > QS [ therefore the sum of the two sides of a triangle is greater than the third side]

In ∆RSP

(RS + SP) > PR [ therefore the sum of the two sides of a triangle is greater than the third side]

2(PQ + QR + RS + SP) > 2(PR + QS) [ adding all the equations]

(PQ + QR + RS + SP) > 2(PR + QS) [ cancelling the common factor 2]

Hence proved

Therefore it is proven that the sum of the four sides of a Quadrilateral exceeds the sum of the diagonals.

Also, Check:

- Sum of any Two Sides is Greater than Twice the Median
- The Sum of any Two Sides of a Triangle is Greater than the Third Side

### FAQs on Sum of the Four Sides of a Quadrilateral Greater than the Sum of the Diagonals

**1. What is the sum of four sides of the quadrilateral?**

A quadrilateral with four sides and four angles. The sum of four angles of a quadrilateral is 360 degrees. We can draw two diagonals to the quadrilateral that forms two triangles.

**2. What is the quadrilateral inequality theorem?**

The sum of any pairs of opposite sides of a convex quadrilateral is always less than the sum of the diagonals.

**3. How many diagonals does a quadrilateral have?**

A quadrilateral can be constructed unambiguously if the lengths of its three sides and two diagonals are provided and if its two adjacent sides and three angles are given.