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Why Are the Angles of Equilateral Triangles Always Equal to 60°?

What are the angles in an equilateral triangle equal to?

Summary

  1. The equals sign with a squiggly line over it means 'is congruent to'
  2. All three sides measure 4, so they are all congruent to each other
  3. The sideways 'v' shape means 'angle'
  4. All of the angles of an equilateral triangle are congruent as well
  5. We put little arcs on the angles to show that they have the same measurement
  6. The tiny circle after 60 means 'degrees'
  7. Since angles 'b' and 'c' measure the same as angle 'a', they all measure 60 degrees

Notes

    1. First, let's verify that we have an equilateral triangle
    1. All of the sides of our triangle measure 4, so we have an equilateral triangle
    1. 'Congruent' just means that all the sides are the same length
    2. Since all our sides measure 4, they are all congruent to each other
    1. The interior angles are the angles formed inside the triangle
    2. 'Congruent' means that all the interior angles have the same measure
    1. The Triangle Sum Theorem says that the interior angles of a triangle always add up to 180 degrees
    2. This is true for ANY triangle!
    3. So we can add up the measurements of the three angles in our triangle and set them equal to 180 degrees
    4. The little 'm' in front of the sideways 'v' means 'the measurement of angle ...'
    5. The tiny circle after 180 means 'degrees'
    1. Remember, since we have an equilateral triangle we know that all the angles are equal
    2. So using the Substitution Property of Equality, we could just substitute the measure of angle 'a' for all three of the angles
    1. The little 'm' in front of the sideways 'v' means 'the measurement of angle ...'
    2. The tiny circle after 180 means 'degrees'
    3. We can just replace the measurement of angles 'b' and 'c' with the measurement of angle 'a', since we know they are all equal
    4. We're adding the same thing 3 times, which is the same as multiplying by 3
    5. So we can rewrite this equation as 3 times the measurement of 'a' instead
    1. The little 'm' in front of the sideways 'v' means 'the measurement of angle ...'
    2. Divide by 3 on both sides to get the measurement of 'a' by itself
    3. The tiny circle after 180 means 'degrees'
    4. We find that the measure of angle 'a' is 60 degrees
    5. This means that the measurements of 'b' and 'c' are also 60 degrees!
    1. This is true for ANY equilateral triangle!
    2. It doesn't matter what the length of the sides are, as long they are all equal, the angles will all measure 60 degrees!
    3. The tiny circle after 60 means 'degrees'