Triangles - Basic Concepts

A closed plane figure requires at least 3 lines. These are called Triangles, “tri” meaning three.

Triangles were one of the earliest shapes studied. This could also be due to the fact that it was both a simple as well as a rigid structure. It is simple because it just needs 3 straight lines to build it.

What is the meaning of a rigid structure? In a triangle, once the 3 sides are fixed, the 3 angles are also fixed. This is not true of any other shape. For example consider a rectangle with certain length & breadth. It can be distorted into a parallelogram with the same length & breadth but with different angles. Hence a rectangle is not rigid. The shape of a triangle cannot be distorted.

This is the reason, triangular structures are used for building larger structures like bridges.

All triangles have 3 sides, 3 angles and 3 vertices. Some of the properties of a triangle are intuitive.

1. At each vertex we can imagine an angle which is inside the triangle and an angle supplementary to it which is outside (exterior) to the triangle.

2. If we stand at one of the vertices, there are 2 ways of reaching any of the other 2 vertices; one directly and the other through the 3rdvertex. This is the rule that “sum of any two sides of a triangle is bigger than the 3rdside”.

3. Imagine a person starting from one of the vertices with a stick in his hand pointing along one of the sides. Let him proceed along this side and when he reaches the other vertex, turn to his left so that the stick again points along the other side which also starts from that vertex. Repeating the same move let him arrive at the starting point. We notice the following

   1. The stick in the hand has turned through a complete angle. This proves that the sum of the exterior angles of a triangle equal a complete angle.

   2. The triangle has 3 vertices and hence 3 pairs of interior & exterior angles. At each vertex the interior & exterior angles are supplementary. Hence the total of the interior & exterior angles of a triangle would be 3 times 180 degrees.

   3. Hence with a little arithmetic manipulation we can see that the sum of the interior angles of a triangle would be 180 degrees.

4. It is intuitively apparent that the sides of a triangle can be arranged in the same order as the measure of the angle opposite to the sides. Hence the shortest side will be opposite the smallest angle & the longest side would be opposite to the biggest angle.

The angle sum property of a triangle can also be arrived by folding a triangle made of paper. All the 3 vertices would meet at a point without any overlap or gap and form a straight angle. (Refer activities in Chapter 25.3)

With the same activity we can see that any triangle can be folded into a rectangle which is half the area of the triangle. This leads to a simple formula for the area of a triangle.

We can see that 3.a would be true for any figure drawn on a plane. Hence the sum of the exterior angles of any polygon would be a complete angle.

We can also intuitively understand that the sum of the angles of a triangle may not total to 180 degrees, if the surface on which the triangle is drawn is curved rather than plane.

1. Let us take an extreme case of a curved surface; that of the surface of the Earth.

2. Let us imagine the 3 lines of the triangle as follows. One vertex at the North Pole. 2 vertices on the equator; one at 0-degree longitude and the other at 90 degrees longitude.

3. If we imagine the triangle drawn on the globe, we can see that all the 3 angles would be 90 degrees. Hence the sum of the angles would be more than 180 degrees!

4. Hence the property that 3 angles of a triangle are 180 degrees also determines that the surface is plane and not curved.

This is the reason the geometry of Euclid dealing with 2D figures is also called Plane Geometry.

Any polygon can be divided into triangles. Hence any property of a polygon can be worked out by working out the properties of a triangle.

Measuring Distance to Stars

Astronomers use a clever math trick that involves creating triangles and applying trigonometry to figure out distances, angles, and heights. There is a Scientific American video which explains this.