Question

Prove angle opposite to longer side is larger​

Answer 1

Answer:

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

Step-by-step explanation:

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:

1. In ∆XYP, ∠XYP = ∠XPY    [XY = XP ]

2. ∠XPY = ∠XZY + ∠PYZ  [ In ∆YPZ, exterior ∠XPY = Sum of interior opposite angles, ∠PZY (=∠XZY) and ∠PYZ. ]

3. Therefore, ∠XPY > ∠XZY.   [ From statement 2]

4. Therefore, ∠XYP > ∠XZY.   [Using statements 1 in 3.]

5. But ∠XYZ > ∠ XYP.   [∠XYZ = ∠XYP + ∠PYZ]

6. Therefore, ∠XYZ > ∠XZY.   [Using statements 5 and 4.]

                      hence proved!