Name: amber
Who is asking: Student
Level: Secondary

In defining the types of triangles, our class was stumped by a question asked by one of the student. Maybe you could help. The definition of an equilateral triangle is a triangle with three congruent sides. The definiton of an isosceles triangle is a triangle with at LEAST two congruent sides. The question is, if an isosceles triangle only requires at Least two of the sides to be congruent, could an equilateral triangle be called an isosceles triangle?

Hi Amber,

I first want to compliment whoever made the definition that you gave me, "An isosceles triangle is a triangle with at LEAST two congruent sides". The dictionaries that I looked in defined isosceles as "having two equal sides", leaving it to the reader to interpret whether or not this means exactly two. With your definition it is explicit that any triangle with at least two congruent sides is isosceles. Thus a triangle with all sides congruent is both isosceles and equilateral.
   Even if the definition were worded "An isosceles triangle is a triangle with two congruent sides" I would say that an equilateral triangle is also an isosceles triangle. An equilateral triangle has two congruent sides, in fact it has three. The definition you gave is better because it is explicit.
   Does your text give a definition of a rectangle? Is a square also a rectangle?


The standard way these things are handled in math is to treat definitions in an inclusive sense. Yes an equilateral triangle is an isosceles triangle - for each of the three possible pairs of sides. That means you can apply the Isosceles Triangle Theorem and conclude that the three angles of an equilateral triangle are also equal!
   A similar thing happens with words like 'square' 'rectangle' and parallelogram.
   A rectangle is a special type of parallelogram - with all angles equal to 90. But it is a parallelogram, and any theorem about parallelograms applies immeditately to a rectangle. (E.g. a theorem about the diagonals bisecting one another).
   Similarly, a square is a special rectangle - but it is a rectangle and a parallelogram.

Walter Whiteley

Euclid Book I, Definition 20 makes it clear that isosceles triangles have exactly two congruent sides (the "legs") and one side of a different length (called the "base"). He distinguishes the three types of triangles: equilateral (3 congruent sides), isosceles (2 congruent sides but not the third), and scalene (no two congruent). On the other hand, since not everybody knows this rule it is wise to remind the reader when the distinction is important by saying things like "an isosceles triangle that is not equilateral", or "a triangle with exactly two congruent sides."


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