Prove that the angles of an equilateral triangle are 60 degrees each.

To prove that the angles of an equilateral triangle are each 60 degrees, we can use some basic geometric principles. Here's a proof:

Let's consider an equilateral triangle ABC, where all three sides are equal in length, and we want to prove that each of its angles measures 60 degrees.

Draw triangle ABC.

Since it's an equilateral triangle, all sides are of equal length. Let's denote the length of each side as "s."

Bisect one of the angles, say angle A, using a straight line segment AD that meets the opposite side BC at point D. This will divide angle A into two equal angles, each measuring "x" degrees.

Similarly, bisect angle B using a straight line segment BE that meets AC at point E, dividing angle B into two equal angles, each measuring "y" degrees.

Now, we have two right triangles, ADE and BEC. In both of these right triangles, we have:

AD = BE (because both are bisectors)
AE = EC (because both sides of the equilateral triangle are equal)
Angle DAE = Angle ECB (each is half of angles A and B, respectively)
By the Side-Angle-Side (SAS) congruence criterion, we can conclude that triangles ADE and BEC are congruent.

Therefore, the remaining angles in these triangles must also be congruent. So, angle AED must be equal to angle BEC, and both measure "y" degrees.

Now, we can consider angle CED, which is the angle at the top of the equilateral triangle.

The sum of the angles in triangle CDE must be 180 degrees (since it's a straight line).

So, we have:

angle CED + angle AED + angle CDE = 180 degrees
y + y + x = 180 degrees

Simplifying:
2y + x = 180 degrees

Now, let's consider angle CED. We know that the sum of the angles in a triangle is 180 degrees.

In triangle CED, we have:

angle CED + angle CDE + angle CDE = 180 degrees
angle CED + y + y = 180 degrees

Simplifying:
angle CED + 2y = 180 degrees

Now, we can equate the expressions for angle CED from steps 11 and 14:
2y + x = angle CED + 2y

The "2y" terms cancel out:
x = angle CED

Now, we have:
x = angle CED

We can substitute the value of "x" from step 3:
x = angle AED

Therefore, angle AED is equal to angle CED, and both are equal to "x" degrees.

Since angles AED and CED are both angles at the top of the equilateral triangle, and they are equal, we can conclude that each of these angles measures "x" degrees.

We have shown that angle AED, angle CED, and angle AED are all equal, so each of them must measure "x" degrees.

Therefore, each angle in the equilateral triangle ABC measures "x" degrees, which means each angle is 60 degrees.

This completes the proof that the angles of an equilateral triangle are each 60 degrees.