# Triangle

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## Meaning

Under construction: fix pictures so that it doesn't look like triangles have to be blue-green

Informally: The word "triangle" is so often taught as early as preschool that it is, for most children, a familiar term by the time they reach elementary school. But the "meaning" learned in preschool tends to be associated so strongly with particular images like these (left to right, the most to least familiar) that there is still new learning for the children to do. The tend not to recognize triangles that are extreme, or in unfamiliar positions, and they might well include as "triangles" figures that resemble their image even though they fail to fit the definition. They need also to see, explicitly the following:

Each of these is a triangle:
None of these is a triangle:

Formal definition: A triangle is a polygon that has three sides. The restriction that it be a polygon is enough to exclude the cases (above) that are not triangles, but a definition alone is too terse for young children, not sufficient for them to build the right concept. Examples and non-examples, like the ones shown above, are necessary before the definition will have sufficient meaning.

## Classification

Triangles are typically classified according it their side lengths or according to their angle measures. When classified according to side lengths, a triangle may be scalene, isosceles, or equilateral. As with sides, there are three classifications of triangles according to angle measures. A triangle is called acute, right, or obtuse according to the measure of its largest angle.

Under construction: Include pictures of each of the possible combinations, maybe a large 'X' for the impossible combinations.

Triangle Classifications
Scalene Isosceles Equilateral
Acute yes yes yes
Right yes yes no
Obtuse yes yes no

Initially, this leads to nine possible classes. Students often expect these classes to be mutually exclusive and non-empty. However, both ideas are incorrect. First, some combinations are not possible. For example, in an equilateral triangle, each angle measures 60 degrees. Hence, every equilateral triangle is an acute triangle and their are no examples of equilateral right triangles or obtuse equilateral triangles.

Second, some classes include others. For example, since every equilateral triangle has three congruent sides, it also has at least two congruent sides. Hence, every equilateral triangle is also an isosceles triangle. In terms of the classification system, the set of (acute) equilateral triangles is a subset of the set of acute isosceles triangles.

## Mathematical background

An important theorem in Euclidean geometry is the Triangle Sum Theorem, which is a special case of the Polygon Sum Theorem.

Triangle Sum Theorem: The sum of the measures of the angles of a triangle is 180 degrees.

Under construction: pictures needed to demonstrate tearing and more formal proof

Students often discover this theorem by measuring the angles of several triangles using a protractor, or by using geometry software. Students can informally suggest this theorem by tearing off two angles of a triangle and aligning them with the third angle of the triangle to form a straight line.

Under construction: (flesh out and illustrate) show a right triangle doubled, to show sum of two acute angles as a right angle; then do the general case by dropping a perpendicular to the longest side from the opposite vertex

In more advanced classes, students learn a proof of the Triangle Sum Theorem that involves drawing an auxiliary line and using properties of parallel lines to identify congruent angles. This reveals a surprising connection between parallel lines and triangles.

• Exterior Angle Theorem
• Third Angle Theorem