Greater Angle has the Greater Side Opposite to It

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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

Greater Angle has the Greater Side Opposite to It

To Prove: XZ > XY

Proof:

            Statement

1. 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 = ∠XZY

2. But ∠XYZ ≠ ∠XZY

3. Again, XZ < XY

⟹ ∠XYZ <
∠XZY

4. But ∠XYZ is not less than ∠XZY.

5. Therefore, XZ is neither equal to nor
less than XY.

6. Therefore, XZ > XY. (Proved)

            Reason

1. Angles opposite to equal sides are
equal.

2. Given, ∠XYZ > ∠XZY

3. 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.

9th Grade Math

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