Types of Triangles Help

Introduction to Types of Triangles

Triangles can be categorized qualitatively (that means according to their qualities or characteristics). Here are the most common character profiles.

Acute Triangle

When each of the three interior angles of a triangle are acute, that triangle is called an acute triangle . In such a triangle, none of the angles measures as much as 90° (π/2 rad). Examples of acute triangles are shown in Fig. 2-9.

Fig. 2-9 . In an acute triangle, all angles measure less than 90° (π/2 rad). 

Obtuse Triangle

When one of the interior angles of a triangle is obtuse, that triangle is called an obtuse triangle . Such a triangle has one obtuse interior angle, that is, one angle that measures more than 90° (π/2 rad). Some examples are shown in Fig. 2-10.

Fig. 2-10 . In an obtuse triangle, one angle measures more than 90° (π/2 rad).

Isosceles Triangle

Suppose we have a triangle with sides S, T , and U , having lengths s, t , and u , respectively. Let x, y , and z be the angles opposite sides S, T , and U , respectively. Suppose any of the following equations hold:

s = t

t = u

s = u

x = y

y = z

x = z

One example of such a situation is shown in Fig. 2-11. This kind of triangle is called an isosceles triangle , and the following logical statements are true:

s = t x = y

t = u y = z

s = u x = z 

Fig. 2-11 . Isosceles triangle.

The double-shafted double arrow ( ) means “if and only if.” It is well to remember this. You should also know that a double-shafted single arrow pointing to the right (⇒) stands for “implies” or “means it is always true that.” When we say s = t x = y , it is logically equivalent to saying s = t = ⇒ x = y and also x = y ⇒ s = t .

Equilateral Triangle

Suppose we have a triangle with sides S, T , and U , having lengths s, t , and u , respectively. Let x, y , and z be the angles opposite sides S, T , and U , respectively. Suppose either of the following are true:

s = t = u

or

x = y = z

Then the triangle is said to be an equilateral triangle (Fig. 2-12), and the following logical statement is valid:

This means that all equilateral triangles have precisely the same shape; they are all directly similar. (They all happen to be inversely similar, too.)

Fig. 2-12 . Equilateral triangle.

Right Triangle

Suppose we have a triangle Δ PQR with sides S, T , and U , having lengths s, t , and u , respectively. If one of the interior angles of this triangle measures 90° (π/2 rad), an angle that is also called a right angle , then the triangle is called a right triangle . In Fig. 2-13, a right triangle is shown in which PRQ is a right angle. The side opposite the right angle is the longest side, and is called the hypotenuse . In Fig. 2-13, this is the side of length u .

Fig. 2-13 . Right angle

Practice problems for these concepts can be found at: Triangle Practice Test.