Shape 

Copyright David & Cynthia Thomas, 2009

Classification & Hierarchical Relationships--Concepts & Connections

Triangle names typically refer to their characteristic features.  For example, triangles having three congruent sides are called equilateral.  Triangles having exactly two congruent sides are called isosceles.  If no two sides of a triangle are congruent, the triangle is called scalene.  If every angle of a triangle has a measure less than 90°, the triangle is called acute.  If one angle has a measure greater than 90°, the triangle is called obtuse.  And if one angle has a measure of exactly 90°, the triangle is called right.

Table 2.3 illustrates several classes of triangles commonly found in elementary mathematics textbooks.  Triangles can be named by either side or angle descriptors or even by both.  For example a scalene acute triangle had no sides the same and all the angles are less than 90°.  In print, these objects are typically identified as equilateral acute, isosceles acute, scalene acute, isosceles obtuse scalene obtuse, isosceles right, and scalene right triangles.  Notice the size of the angle is relative to the opposite side ( ie. The largest angle is opposite the longest side. Angles that are congruent are opposite sides that are congruent).   Note: Two of the cells of the table are blank, corresponding to equilateral obtuse and equilateral right triangles.  Explain why these triangles cannot exist.

	Acute	Obtuse	Right
Scalene	Scalene Acute	Scalene Obtuse	Scalene Right
Isosceles	Isosceles Acute	Isosceles Obtuse	Isosceles Right
Equilateral	Equilateral Acute

Table 2.3: Triangle Names

A number of right triangles are commonly designated by specifying their angle or side measurements.  Table 2.4 presents four right triangles commonly found in elementary textbooks.  Note that in two of the triangles the angles are labeled and in the other two triangles the sides are labeled.  The triangles containing 90° angles are, by definition, right triangles.  With regard to the other two triangles, we conclude that they are right triangles because their sides satisfy the Pythagorean relationship, a² + b² = c².    

Angle Specifications	Side Specifications
30°- 60° - 90°	3 – 4 - 5
45° - 45° - 90°	5 – 12 – 13

Table 2.4: Right Triangle Specifications

Relationships between different classes or sets of triangles are often illustrated using tree diagrams (See Figure 2.25).  Reading from the top down, the set of all triangles is partitioned by angle descriptors into three disjoint (i.e., they have no triangles in common) subsets: Right triangles; obtuse triangles; and acute triangles.  These subsets are then further subdivided into equilateral, isosceles, and acute triangles. 

Figure 2.25: Triangle Relationships

In this representation, a triangle in any subset is understood to be a member of all sets “above” it in the diagram.  For example, every equilateral triangle is also an acute triangle.  Also, no right triangle is acute.  Learning to express relationships of this sort using terms like all, some, and no (or none) is a critical aspect of mathematical dialogue.