# The Sum of any Two Sides of a Triangle is Greater than the Third Side

Here we will prove that the sum of any two sides of a triangle is greater than the third side.

Given: XYZ is a triangle.

To Prove: (XY + XZ) > YZ, (YZ + XZ) > XY and (XY + YZ) > XZ

Construction: Produce YX to P such that XP = XZ. Join P and Z.

 Statement1. ∠XZP = ∠XPZ.2. ∠YZP > ∠XZP.3. Therefore, ∠YZP > ∠XPZ.4. ∠YZP > ∠YPZ.5. In ∆YZP,  YP > YZ.6. (YX + XP) > YZ. 7. (YX + XZ) > YZ. (Proved) Reason1. XP = XZ.2. ∠YZP = ∠YZX + ∠XZP.3. From 1 and 2.4. From 3.5. Greater angle has greater side opposite to it.6. YP = YX + XP 7. XP = XZ

Similarly, it can be shown that (YZ + XZ) >XY and (XY + YZ) > XZ.

Corollary: In a triangle, the difference of the lengths of any two sides is less than the third side.

Proof: In a ∆XYZ, according to the above theorem (XY + XZ) > YZ and (XY + YZ) > XZ.

Therefore, XY > (YZ - XZ) and XY > (XZ - YZ).

Therefore, XY > difference of XZ and YZ.

Note: Three given lengths can be sides of a triangle if the sum of two smaller lengths greater than the greatest length.

For example: 2 cm, 5 cm and 4 cm can be the lengths of three sides of a triangle (since, 2 + 4 = 6 > 5). But 2 cm, 6.5 cm and 4 cm cannot be the lengths of three sides of a triangle (since, 2 + 4 ≯ 6.5).

From The Sum of any Two Sides of a Triangle is Greater than the Third Side to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Constructing a Line Segment |Construction of Line Segment|Constructing

Aug 14, 24 09:52 AM

We will discuss here about constructing a line segment. We know how to draw a line segment of a certain length. Suppose we want to draw a line segment of 4.5 cm length.

2. ### Construction of Perpendicular Lines by Using a Protractor, Set-square

Aug 14, 24 02:39 AM

Construction of perpendicular lines by using a protractor is discussed here. To construct a perpendicular to a given line l at a given point A on it, we need to follow the given procedure

3. ### Construction of a Circle | Working Rules | Step-by-step Explanation |

Aug 13, 24 01:27 AM

Construction of a Circle when the length of its Radius is given. Working Rules | Step I: Open the compass such that its pointer be put on initial point (i.e. O) of ruler / scale and the pencil-end be…

4. ### Practical Geometry | Ruler | Set-Squares | Protractor |Compass|Divider

Aug 12, 24 03:20 PM

In practical geometry, we study geometrical constructions. The word 'construction' in geometry is used for drawing a correct and accurate figure from the given measurements. In this chapter, we shall…