Math CentralQuandaries & Queries


Question from Jane, a teacher:

If all sides of a triangle are the same length and all angles are 60 degrees, can the triangle be called both an acute triangle and an equilateral triangle?

The answer is YES.

In mathematics, the usual way to work with classifications / definitions is as 'inclusive' properties.
As long an object retains the key property, it remains in the category, even if some extra properties come up.

So a square IS still a rectangle (still has four right angles!). Both are also parallelograms. One way to see that this is useful, is to realize that any special properties from the larger class:
e.g. diagonals bisect in a parallelogram) still apply to the special cases (e.g. a square or a rectangle).
e.g. the formula for the area of a rectangle still works for a square,

A second way to 'see' this is to consider constructions in a dynamic geometry program (like Geometer's SketchPad). If you use the key positive properties to define what a parallelogram is, and then start moving things around, one of the images that appears is a rectangle, maybe even a square. You don't say that the construction 'failed'! It just was inclusive of all the objects you found, and included the rectangle and square as possible examples.

So - back to triangles: An Equilateral Triangle is still Isosceles. It does have two equal sides (and more). It also has two equal angles (and more)!

Acute is a somewhat different property but clearly all angles of an equilateral triangle are less than 90 degrees. When possible we just find it easier to think, and problem solve, in mathematics, with inclusive definitions.


About Math Central


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS