A polygon bounded by three line segments is known as the Triangle. It is the smallest possible polygon. A triangle has three sides, three vertices, and three interior angles.

**Triangle Inequality Theorem**

The triangle inequality theorem describes the relationship between the three sides of a triangle. According to this theorem, for any triangle, the sum of lengths of two sides is always greater than the third side. In other words, this theorem specifies that the shortest distance between two distinct points is always a straight line.

Consider a ∆ABC as shown below, with a, b and c as the side lengths.

The triangle inequality theorem states that:

a < b + c,

b < a + c,

c < a + b

In any triangle, the shortest distance from any vertex to the opposite side is the Perpendicular. In fig. below, XP is the shortest line segment from vertex X to side YZ.

**Proof for Triangle Inequality Theorem –**

**To Prove –**

|BC|< |AB| + |AC|

Construction: Consider a ∆ABC. Extend the side AC to a point D such that AD = AB as shown in the fig. below.

S. No |
Statement |
Reason |

1. | |CD|= |AC| + |AD| | From figure 3 |

2. | |CD|= |AC| + |AB| | AB = AD, ∆ADB is an isosceles triangle |

3. | ∠DBA <∠DBC | Since ∠DBC = ∠DBA+∠ABD |

4. | ∠ADB<∠DBC | ∆ADB is an isosceles triangle and ∠ADB = ∠DBA |

5. | |BC|<|CD| | Side opposite to larger angle is larger in length |

6. | |BC|<|AC| + |AB| | From statements 3 and 4 |

Thus, we can conclude that the sum of two sides of a triangle is greater than the third side.

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