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.
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Statement 1. ∠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) |
Reason 1. 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).
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