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# Types of Triangles Help

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## 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 = ys = 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.

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