Triangles – Congruency & Similarity

Congruency & Similarity

These again are two important properties connected with plane figures. Congruency relates to shape & size and similarity relates only to the shape. If tow figures are identical in all respects, they are said to be congruent.

Two triangles are congruent if they have identical sides and angles. They can completely overlap each other. In practice, to check if 2 triangular shapes are congruent, we can just place then overlapping and see if the fit is perfect.

But in many real-life situations, we may not be able to physically overlap the triangles. One example could be for comparing the sizes of 2 triangular plots which are located at different locations. Hence Greeks studied triangle in the abstract and derived rules to check if triangles occurring in different parts of their diagrams where congruent or not.

Congruency can be interpreted in a simpler way. Given 3 measures of any triangle, out of the 3 sides & 3 angles, if only 1 triangle can be drawn then it is a valid condition for congruency.

Similarity

While congruency relates to both the size and shape of figures, similarity is only concerned with shape. It is an aspect which allows us to identify a friend from his passport sized photograph which is several times smaller. It has to do with the proportion of various dimensions of a face in the photograph being in certain proportion.

Greeks discovered that 2 triangles will look similar if their corresponding sides are in the same ratio i.e proportional. In the case of triangles, this translates into a simpler property that the corresponding angles of similar triangles are equal. One of the earliest Greek philosophers, Thales is supposed to have worked out the height of an Egyptian pyramid by applying the law of similarity to the length of its shadow.

It is never emphasized in the curriculum that similarity is a concept which applies to all shapes and that the fundamental property of similarity relates to ratio of the corresponding sides. And that it just happens that in the case of triangles it turns out that in similar triangles, the corresponding angles are equal!

Students assume that only triangles have the property of similarity and that angles are fundamental to the idea of similarity. We will study this issue in detail when we study quadrilaterals.

Triangles & Mapping

If the side of a triangle and the 2 angles at its ends are known, the 3rdpoint is uniquely determined. This is the meaning of saying that the triangle is a rigid figure. If any of its angles are changed, the sides also change. Polygons with more than 3 sides do not have this property.

This is also called the ASA condition of congruence. Angles can be more easily measured, even over very long distances. This property is widely used in mapping the surface of the Earth and to carry out surveys.

Starting from a fixed base which is accurately measured, a 3rdlocation (usually a prominent landmark) is located by the 2 angles subtended by the location at the endpoints of the 2 ends of the line. This process, called Triangulation, is repeated to locate more points, until the entire area is covered.

The survey of India, which started in 1800, which also identified Mount Everest as the tallest peak in the world, is a story which needs to be taught to all students in Indian schools. It started from a hill in Madras city and used iron chains, almost a mile long to mark other points and measure the angles formed by the triangle.

The entire story has been narrated in "The Great Arc: The Dramatic Tale of How India Was Mapped and Everest Was Named " by John Keay.