Three Angles of an Equilateral Triangle are Equal

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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.

Three Angles of an Equilateral Triangle

To prove: XY = YZ = ZX.


Proof:

   Statement

1. XY = ZX.

2. XY = YZ.

3. XY = YZ = ZX.

(Proved)

               Reason

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.

Bisector of an Isosceles Triangle

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.

Three Axes of Symmetry of an Equilateral Triangle

An isosceles triangle has one axis of symmetry while the equilateral ∆ABC has three axes of symmetry, AP, BQ and CR.

9th Grade Math

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