Here we will prove that if the three angles of a triangle are equal, it is an equilateral triangle.
Given: In ∆XYZ, ∠YXZ = ∠XYZ = ∠XZY.
To prove: XY = YZ = ZX.
1. XY = ZX.
2. XY = YZ.
3. XY = YZ = ZX.
1. Sides opposite to equal angles ∠XZY and ∠XYZ.
2. Sides opposite to equal angles ∠XZY and ∠ZXY.
3. from statement 1 and 2.
Note: In the adjoining figure, ∆XYZ is an isosceles triangle in which XY = XZ. XM is the bisector of ∠YXZ.
If the triangle is folded along the line XM, the side XY will fall along XZ because ∠YXM = ∠ZXM, and Y will coincide with Z as XY = XZ. Thus, YM will coincide with ZM. This shows ∠XYZ = ∠XZY.
Also, ∠XMY = ∠XMZ = 90°. ∆XYM coincides with ∆XZM. So, ∆XYZ is said to be symmetrical about the line XM. The line XM is called the axis of symmetry.
An isosceles triangle has one axis of symmetry while the equilateral ∆ABC has three axes of symmetry, AP, BQ and CR.