The Equilateral Triangle Theorem: Explained and Proven – All Angles of an Equilateral Triangle are 60 Degrees

Equilateral Triangle Theorem

The Equilateral Triangle Theorem states that if a triangle is equilateral, then all angles of the triangle are congruent (i

The Equilateral Triangle Theorem states that if a triangle is equilateral, then all angles of the triangle are congruent (i.e., equal) and each angle measures 60 degrees.

To prove this theorem, we can start by considering an equilateral triangle with sides of length ‘s’. Let’s label the vertices of the triangle as A, B, and C.

Since the triangle is equilateral, all sides are congruent, and thus side AB = BC = AC = s.

Now, let’s examine the angles of the equilateral triangle. We want to show that each angle, ∠A, ∠B, and ∠C, measures 60 degrees.

To do this, we can divide the triangle into two congruent right-angled triangles by drawing an altitude from any vertex (say, vertex A) to the midpoint of the opposite side, which we can label as point D.

Since AD is drawn perpendicular to BC, AD bisects BC. Therefore, BD = DC = s/2.

Now, let’s consider the right-angled triangle ABD. We have the following information:

– AD = s/2 (because it is the height of the equilateral triangle)
– BD = s/2 (because BC = AC = s)
– AB = s (because it is a side of the equilateral triangle)

Using the Pythagorean theorem in triangle ABD, we can determine the length of AD:

(AD)^2 = (AB)^2 – (BD)^2
(s/2)^2 = s^2 – (s/2)^2
s^2/4 = (4s^2)/4 – s^2/4
s^2/4 = (4s^2 – s^2)/4
s^2/4 = 3s^2/4
s^2 = 3s^2
s^2 – 3s^2 = 0
-2s^2 = 0

From this equation, we can see that s^2 = 0, which means s = 0. However, since s represents the length of a side of the equilateral triangle, it cannot be zero. Therefore, -2s^2 = 0 cannot be satisfied.

Therefore, the only solution to the equation is a degenerate triangle, which is not possible in the context of an equilateral triangle.

Hence, we conclude that our assumption that s = 0 is incorrect, and therefore s cannot equal 0. Consequently, the only possible value for s is a positive number.

In other words, we have proven that the line AD is never zero, which means that the triangle ABD is always a valid right-angled triangle. Since the angles of a right-angled triangle are fixed (with one angle being 90 degrees), we can conclude that ∠BAD is also 90 degrees.

Since we have shown that ∠BAD = 90 degrees and the equilateral triangle has three congruent sides, it follows that each angle of the equilateral triangle measures 60 degrees.

Hence, the Equilateral Triangle Theorem states that if a triangle is equilateral, then all angles of the triangle are congruent and each angle measures 60 degrees.

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