**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.

**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.

**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*

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.)

**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* .

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