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.
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.
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° = \(\frac{180° - y°}{2}\).
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