The Three Angles of an Equilateral Triangle are Equal

Here we will prove that the three angles of an equilateral triangle are equal.

Given: PQR is an equilateral triangle.

The Three Angles of an Equilateral Triangle are Equal

To prove: ∠QPR = ∠PQR = ∠ PRQ.





Proof:

          Statement

1. ∠QPR = ∠PQR

2. ∠PQR = ∠ PRQ.

3. ∠QPR = ∠PQR = ∠ PRQ. (Proved).

          Reason

1. Angles opposite to equal sides QR and PR.

2. Angles opposite to equal sides PR and PQ.

3. From statement 1 and 2.

Note:

1. In the equilateral ∆PQR, let ∠PQR = ∠PRQ = ∠RPQ = x°. Therefore, 3x° = 180° as the sum of the three angles of a triangle is 180°.

Therefore, x° = \(\frac{180°}{3}\)

⟹ x° = 60°.

Thus, each angle of an equilateral triangle is 60°.


2. If one angle of an isosceles triangle is given, the other two can be easily found out.

One Angle of an Isosceles Triangle

In the given figure, PQ = PR.

Therefore, ∠PQR = ∠PRQ = x° (suppose).

Let ∠RPQ = y°

Thus, y° + 2x° = 180°, from which we get

y° = 180° - 2x°

and x° = \(\frac{180° - y°}{2}\).








9th Grade Math

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