**Introduction to Triangles**

If you ever took a course in plane geometry, you remember triangles. Do you recall being forced to learn formal proofs about them? We won’t go through proofs here, but important facts about triangles are worth stating. If this is the first time you’ve worked with triangles, you should find most of the information in this chapter intuitively easy to grasp.

In mathematics, it’s essential to know exactly what one is talking about, without any “loopholes” or ambiguities. This is why there are formal definitions for almost everything (except primitives such as the point and the line).

**What Is A Triangle?**

First, let’s define what a triangle is, so we will not make the mistake of calling something a triangle when it really isn’t. A triangle is a set of three line segments, joined pairwise at their end points, and including those end points. The three points must not be collinear; that is, they must not all lie on the same straight line. For our purposes, we assume that the universe in which we define the triangle is Euclidean (not “warped” like the space around a black hole). In such an ideal universe, the shortest distance between any two points is defined by the straight line segment connecting those two points.

**Vertices**

Figure 2-1 shows three points, called *A, B* , and *C* , connected by line segments to form a triangle. The points are called the *vertices* of the triangle. Often, other uppercase letters are used to denote the vertices of a triangle. For example, *P, Q* , and *R* are common choices.

**Naming**

The triangle in Fig. 2-1 can be called, as you might guess, “triangle *ABC* .” In geometry, it is customary to use a little triangle symbol (Δ) in place of the word “triangle.” This symbol is actually the uppercase Greek letter delta. Fig. 2-1 illustrates a triangle that we can call Δ *ABC* .

**Sides**

The sides of the triangle in Fig. 2-1 are named according to their end points. Thus, Δ *ABC* has three sides: line segment *AB* , line segment *BC* , and line segment *CA* . There are other ways of naming the sides, but as long as there is no confusion, we can call them just about anything.

**Interior Angles**

Each vertex of a triangle is associated with an *interior angle* , which always measures more than 0° (0 rad) but less than 180° (π rad). In Fig. 2-1, the interior angles are denoted *x* , *y* , and *z* . Sometimes, italic lowercase Greek letters are used instead. Theta (pronounced “THAY-tuh”) is a popular choice. It looks like a leaning numeral zero with a dash across it (θ). Subscripts can be used to denote the interior angles of a triangle, for example, *θ* _{a} , *θ* _{b} , and *θ* _{c} for the interior angles at vertices *A, B* , and *C* , respectively.

**Similar Triangles**

Two triangles are *directly similar* if and only if they have the same proportions in the same rotational sense. This means that one triangle is an enlarged and/or rotated copy of the other. Some examples of similar triangles are shown in Fig. 2-2. If you take any one of the triangles, enlarge it or reduce it uniformly and rotate it clockwise or counterclockwise to the correct extent, you can place it exactly over any of the other triangles. Two triangles are not directly similar if it is necessary to flip one of the triangles over, in addition to changing its size and rotating it, in order to be able to place it over the other.

Two triangles are *inversely similar* if and only if they are directly similar when considered in the opposite rotational sense. In simpler terms, they are inversely similar if and only if the mirror image of one is directly similar to the other.

If there are two triangles Δ *ABC* and Δ *DEF* that are directly similar, we can symbolize this by writing Δ *ABC* ~ Δ *DEF* . The direct similarity symbol looks like a wavy minus sign. If the triangles Δ *ABC* and Δ *DEF* are inversely similar, the situation is more complicated because there are three ways this can happen. Here they are:

- Points
*D*and*E*are transposed, so Δ*ABC*˜ Δ*EDF* - Points
*E*and*F*are transposed, so Δ*ABC*˜ Δ*DFE* - Points
*D*and*F*are transposed, so Δ*ABC*˜ Δ*FED*

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