This document defines and classifies triangles based on the lengths of their sides and the measures of their interior angles. It explains that triangles can be equilateral, isosceles, or scalene based on whether their sides are all equal, two sides are equal, or all sides are unequal. Triangles can also be right, obtuse, or acute based on whether they have a 90 degree angle, an angle over 90 degrees, or all angles under 90 degrees. The document provides examples and diagrams to illustrate different types of triangles.

2.  Euler diagram of types of triangles, using
the definition that isosceles triangles
have at least 2 equal sides, i.e. equilateral
triangles are isosceles.

3.  Triangles can be classified according
to the relative lengths of their sides:
 In an equilateral triangle all sides
have the same length. An equilateral
triangle is also a regular poligon with
all angles measuring 60°.

4.  In an isosceles triangle, two sides are equal
in length. An isosceles triangle also has two
angles of the same measure; namely, the
angles opposite to the two sides of the same
length; this fact is the content of
the Isosceles triangle theorem. Some
mathematicians define an isosceles triangle to
have exactly two equal sides, whereas others
define an isosceles triangle as one with at
least two equal sides. The latter definition
would make all equilateral triangles isosceles
triangles. The 45–45–90 Right Triangle, which
appears in the Tetrakis square tiling, is
isosceles.

5.  In a scalene triangle, all sides are unequal.[4] The
three angles are also all different in measure.
Some (but not all) scalene triangles are also right
triangles.
In diagrams representing triangles (and other
geometric figures), "tick" marks along the sides
are used to denote sides of equal lengths – the
equilateral triangle has tick marks on all 3 sides,
the isosceles on 2 sides. The scalene has single,
double, and triple tick marks, indicating that no
sides are equal. Similarly, arcs on the inside of
the vertices are used to indicate equal angles.
The equilateral triangle indicates all 3 angles are
equal; the isosceles shows 2 identical angles. The
scalene indicates by 1, 2, and 3 arcs that no
angles are equal.

6.  Triangles can also be classified according to their internal
angles, measured here in degrees.
 A right triangle (or right-angled triangle, formerly called
a rectangled triangle) has one of its interior angles measuring
90° (a right angle). The side opposite to the right angle is
the hypotenuse; it is the longest side of the right triangle.
The other two sides are called
the legs or catheti (singular: cathetus) of the triangle. Right
triangles obey the Pythagorean Theorem: the sum of the
squares of the lengths of the two legs is equal to the square
of the length of the hypotenuse: a2 + b2 = c2,
where a and b are the lengths of the legs and c is the length
of the hypotenuse. Special right triangles are right triangles
with additional properties that make calculations involving them
easier. One of the two most famous is the 3–4–5 right
triangle, where 32 + 42 = 52. In this situation, 3, 4, and 5
are a Pythagorean Triple. The other one is an isosceles
triangle that has 2 angles that each measure 45 degrees.

7.  Triangles that do not have an angle that
measures 90° are called oblique triangles.
 A triangle that has all interior angles measuring
less than 90° is an acute triangle or acute-
angled triangle.
 A triangle that has one angle that measures more
than 90° is an obtuse triangle or obtuse-angled
triangle.
 A "triangle" with an interior angle of 180°
(and collinear vertices) is degenerate.
 A triangle that has two angles with the same
measure also has two sides with the same length,
and therefore it is an isosceles triangle. It
follows that in a triangle where all angles have
the same measure, all three sides have the same
length, and such a triangle is therefore
equilateral.

9. Realizzato da:
 Roberta Laneve
 Aurora Petrera
 Sharon Potenza
 Clorinda Stasolla
 Flavia Tropeano