Sides Opposite to the Equal Angles of a Triangle are Equal

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Here we will prove that the sides opposite to the equal
angles of a triangle are equal.

Given: In ∆ABC, ∠XYZ = ∠XZY.

Sides Opposite to the Equal Angles of a Triangle are Equal

To prove: XY = XZ.


Construction: Draw the bisector XM of ∠YXZ so that it meets YZ
at M.

Proof:

          Statement

1. In ∆XYM and ∆XZM,

(i) ∠XYM = XZM

(ii) ∠YXM = ∠ZXM

(iii) XM = XM.

2. ∆XYM ≅ ∆XZM

3. XY = XZ. (Proved)

          Reason

1.

(i) Given.

(ii) XM bisects ∠YXZ.

(iii) Common side.

2. By AAS criterion.

3. CPCTC.

9th Grade Math

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