Greater Side has the Greater Angle Opposite to it
Here we will prove that if two sides of a triangle are unequal, the greater side has the greater angle opposite to it.
Given: In ∆XYZ, XZ > XY
To prove: ∠XYZ > ∠XZY.
Construction: From XZ, cut off XP such that XP equals XY. Join Y and P.
Proof:
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Statement 1. In ∆XYP, ∠XYP = ∠XPY 2. ∠XPY = ∠XZY + ∠PYZ |
Reason 1. XY = XP 2. In ∆YPZ, exterior ∠XPY = Sum of interior opposite angles, ∠PZY (=∠XZY) and ∠PYZ. |
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3. Therefore, ∠XPY > ∠XZY. 4. Therefore, ∠XYP > ∠XZY. 5. But ∠XYZ > ∠ XYP. 6. Therefore, ∠XYZ > ∠XZY. (Proved) |
3. From statement 2. 4. Using statements 1 in 3. 5. ∠XYZ = ∠XYP + ∠PYZ 6. Using statements 5 and 4. |
Note: The angle opposite to the greater side in a triangle is the greatest in measure.
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