Triangle Definitions Help (page 3)
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.
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.
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 .
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.
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.
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
There is disagreement in the literature about the exact meaning of the terms congruence and congruent when describing geometric figures in a plane. Some texts say two objects in a plane are congruent if and only if one can be placed exactly over the other after a rigid transformation (rotating it or moving it around, but not flipping it over). Other texts define congruence to allow flipping over, as well as rotation and motion. Let’s stay away from that confusion, and make two definitions.
Two triangles exhibit direct congruence (they are directly congruent ) if and only if they are directly similar, and the corresponding sides have the same lengths. Some examples are shown in Fig. 2-3. If you take one of the triangles and rotate it clockwise or counterclockwise to the correct extent, you can “paste” it precisely over any of the other triangles. Rotation and motion are allowed, but flipping over, also called mirroring , is forbidden. In general, triangles are not directly congruent if you must flip one of them over, in addition to rotating it, in order to be able to place it over the other.
Two triangles exhibit inverse congruence (they are inversely congruent ) if and only if they are inversely similar, and they are also the same size. Rotation and motion are allowed, and mirroring is actually required.
If there are two triangles Δ ABC and Δ DEF that are directly congruent, we can symbolize this by writing Δ ABC ≅ Δ DEF . The direct congruence symbol is an equals sign with a direct similarity symbol on top. If the triangles Δ ABC and Δ DEF are inversely congruent, the same situation arises as is the case with inverse similarity. Three possibilities exist:
- 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
Facts About Congruent Triangles
Direct Congruence Facts
Here are two important things you should remember about triangles that are directly congruent.
Direct Congruence Fact 1: Corresponding Sides
- If two triangles are directly congruent, then their corresponding sides have equal lengths as you proceed around both triangles in the same direction. The converse of this is also true. If two triangles have corresponding sides with equal lengths as you proceed around them both in the same direction, then the two triangles are directly congruent.
Direct Congruence Fact 2: Corresponding Interior Angles
- If two triangles are directly congruent, then their corresponding interior angles (that is, the interior angles opposite the corresponding sides) have equal measures as you proceed around both triangles in the same direction. The converse of this is not necessarily true. It is possible for two triangles to have corresponding interior angles with equal measures when you proceed around them both in the same direction, and yet the two triangles are not directly congruent.
Inverse Congruence Facts
Here are two “mirror images” of the facts just stated. They concern triangles that are inversely congruent. The wording is almost (but not quite) the same!
Inverse Congruence Fact 1: Corresponding Sides
- If two triangles are inversely congruent, then their corresponding sides have equal lengths as you proceed around the triangles in opposite directions. The converse of this is also true. If two triangles have corresponding sides with equal lengths as you proceed around them in opposite directions, then the two triangles are inversely congruent.
Inverse Congruence Fact 2: Corresponding Interior Angles
- If two triangles are inversely congruent, then their corresponding interior angles have equal measures as you proceed around the triangles in opposite directions. The converse of this is not necessarily true. It is possible for two triangles to have corresponding interior angles with equal measures as you proceed around them in opposite directions, and yet the two triangles are not inversely congruent.
Let P, Q , and R be three distinct points that do not all lie on the same straight line. Then the following statements are true (Fig. 2-4):
- P, Q , and R lie at the vertices of some triangle T
- T is the only triangle having vertices P, Q , and R
Triangle Definition Practice Problems
Suppose you have a perfectly rectangular field surrounded by four straight lengths of fence. You build a straight fence diagonally across this field, so the diagonal fence divides the field into two triangles. Are these triangles directly congruent? If they are not congruent, are they directly similar?
It helps to draw a diagram of this situation. If you do this, you can see that the two triangles are directly congruent. Consider the theoretical images of the triangles (which, unlike the fences, you can move around in your imagination). You can rotate one of these theoretical triangles exactly 180° (π rad), either clockwise or counterclockwise, and move it a short distance upward and to the side, and it will fit exactly over the other one.
Suppose you have a telescope equipped with a camera. You focus on a distant, triangular sign and take a photograph of it. Then you double the magnification of the telescope and, making sure the whole sign fits into the field of view of the camera, you take another photograph. When you get the photos developed, you see triangles in each photograph. Are these triangles directly congruent? If not, are they directly similar?
In the photos, one triangle looks larger than the other. But unless there is something wrong with the telescope, or you use a star diagonal when taking one photograph and not when taking the other (a star diagonal renders an image laterally inverted), the two triangle images have the same shape in the same rotational sense. They are not directly congruent, but they are directly similar.
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