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.

**Proof:**

1. XY = ZX. 2. XY = YZ. 3. XY = YZ = ZX. (Proved) |
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.

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