Greater Angle has the Greater Side Opposite to It

Here we will prove that if two angles of a triangle are unequal, the greater angle has the greater side opposite to it.

Given: In ∆XYZ, ∠XYZ > ∠XZY

To Prove: XZ > XY

Proof:

 Statement1. Let us assume that XZ is not greater than XY. Then XZ must be either equal to or less than XY. Let XZ = XY ⟹ ∠XYZ = ∠XZY2. But ∠XYZ ≠ ∠XZY3. Again, XZ < XY⟹ ∠XYZ < ∠XZY4. But ∠XYZ is not less than ∠XZY.5. Therefore, XZ is neither equal to nor less than XY. 6. Therefore, XZ > XY. (Proved) Reason1. Angles opposite to equal sides are equal.2. Given, ∠XYZ > ∠XZY3. The greater side of a triangle has the greater angle opposite to it.4. Given that ∠XYZ > ∠XZY.5. Both the assumptions are leading to contradictions. 6.  From statement 5.

Note: The side opposite to the greatest angle in a triangle is the longest.

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