Understanding Triangles

It's the shape of coniferous trees and peaks in rugged mountain ranges. Wherever it appears in nature, the triangle grabs the imagination and demands a second look. It's no wonder the triangle forced itself into our consciousness more than 2300 years ago and has nestled in the curriculum of the educated for more than 2000 years. This article discusses triangles and some of their properties.

CONTENTS

1. Where do triangles exist
2. Triangles in a plane
3. Triangles on a sphere
4. Triangles in a finite world
5. Summary and reflections

Section 1 WHERE DO TRIANGLES EXIST
All things exists somewhere. Mathematicians call the place where triangles exist, a space. In this discussion, the first space we consider is a plane. To visualize a plane, think of a flat unbounded tabletop that extends without end in all directions. As a plane extends in all directions without bound, it has infinitely many points.

Section 2 TRIANGLES IN A PLANE
Let A, B and C be the names of three points that are not on a single line in a plane. Connect each pair with the shortest path between the two points. The figure formed by three connecting paths is a triangle. The shortest path between two points in a plane is a line segment. A straightedge can be used to make a line segment.

The connecting line segments of the triangle are called sides of the triangle. They are named with the points they connect. Each of the three points is called a vertex of the triangle. So, the triangle formed by the points A, B, and C has:
• sides: AB, AC and BC and
• vertices: A, B and C.

At each vertex, the two line segments meet and form an angle. The angle of a triangle takes the name of the vertex where the angle is formed. Thus a triangle has three angles: Angle A, Angle B and Angles C.

Some angles have wider openings than others. The unit of measure for an angle is a degree. In the plane, the sum of the three angle measures of a triangle is always 180 degrees.

Triangles are determined to be alike by:
• the measures of their sides or
• the measure of their angles.

CONGRUENT TRIANGLES
If the measures of three sides of one triangle are equal to the measures of three sides of a second triangle, the triangles are called congruent.

Congruent triangles can be placed one upon another while appearing as one triangle. This graphic description of congruency asserts there is only one triangle for a set of three measures of those sides. If the triangle is called the standard, then the triangles congruent to it are copies having different positions and different orientations within the plane.

SIMILAR TRIANGLES
If the angle measures of one triangle are the equal to the angle measures of a second triangle, the two triangles are called similar. The sides of similar triangles are proportional.

For instance: If the points A, B, C and D, E, F form similar triangles and side AB measures 5 inches while side DE measures 2 inches, then the ratio of side BC to side EF is 5 to 2. Likewise, the ratio of side AC to side DF is 5 to 2. Unlike congruency, a triangle with a given set of angle measures has infinitely many triangles similar to it, thus making similar triangles useful to map landscapes.

To map a landscape, divide the landscape into triangles and accurately measure all angles. An accurate map is created when the ratio of a side in a landscape triangle to its corresponding side in the drawing is maintained throughout the map.

Finally, congruent triangles are similar triangles but similar triangles are necessarily congruent.

Section 3 TRIANGLES ON A SPHERE
In this discussion, the space is the surface of a sphere centered at a point called, O. Think of a smooth ball. This space is an approximate model for the surface of the earth.

A Line on the Surface of the Sphere
Let A and B be the name of two points on the sphere. The line containing A and B on the sphere is the circle with center O containing A and B. To visualize the line, think about a plane passing through the center of the sphere and the points A and B. The intersection of the sphere and the plane is a circle. O is the center of the circle. Since A and B are on the circle, they divide this circle into two arcs. The line segment from A to B is the shorter arc joining A and B.

For any three points on the sphere that are not on a line, connect each with a line segment on the sphere. These three line segments form a triangle. As before, a triangle has three sides and three angles. Unlike the theory in the plane, the sum of the angle measure of a triangle can be greater than 180°. For example, consider the North Pole and two points on the equator separated by one-quarter of the length of the equator. These three points form a triangle. Each angle of the triangle is 90°.

In this space, there are infinitely many pairs of points connected by two shortest paths. Consider the North and South Poles. There are two different shortest line segments joining them. This is true for all endpoints of a diameter.

Section 4 TRIANGLES IN A FINITE WORLD
A chessboard models the space. The points of the space are the cells of the chessboard. Thus, the space has 64 points.
As before, a triangle is the three shortest paths joining three points that are not on a line.

In order to measure length in this space, define a cell to be one unit from each cell adjacent to it.

In order to identify points in this space, denote the cell in the r th row and c th column by (c,r) where c gives the column's position from the left of the chessboard and r gives the row's position from the bottom of the chessboard. The cell in the lower left hand corner is (1,1), the cell in the upper left hand corner is (1,8) and (8,8) is the cell in the upper right hand corner, etc.

Some Interesting Triangles
Let A(1,1) and B(1,8) be the end cells of the left most vertical column. If C is the cell (4,5) then A, B, and C are not on a line, so they form a triangle. Consider the following two shortest line segments joining A and B:
a. the points of the left most column and
b. the points:(1,1), (2,2), (3,3), (4,4), (3,5), (2,6), (1,7), and (1,8).

The paths in (a) and (b) have the same length. Since they join A and B, they are the shortest paths joining A and B. Thus the triangle with sides AC, BC and the points of (a); and the triangle with sides AC, BC and the points of (b) are congruent triangles. Unlike in the plane, these congruent triangles do not aligned one on the other and appear as one triangle.

Section 5 SUMMARY AND REFLECTIONS
This treatment discusses triangles in an unbounded infinite space, a plane; in a bounded infinite space, a sphere; and finally in a bounded finite space modeled by a chessboard.

In each space, a triangle was defined the same way. Different spaces presented different truths about the triangle. In 1826, this revelation shocked the mathematical community that previously thought geometric statements as absolute truth. Thus mathematicians were forced to realize that assumptions about a space affect the properties of triangles in the space, thereby forcing a redefinition of the scope of mathematical truth.

As social scientists posit the effect of an environment upon its inhabitants, one can assert a similar statement for mathematics and the hard sciences.