Question

Prove the transitive property of similarity: if A~B and B~C, then A~C.

Answer 1

The transitive property states that if a= b and b= c, then a=c. In algebra the transitive property is used to help solve problems. If there are two different expressions which both equal the same thing, then we can use the transitive property to help connect the two different expressions.  For example, if x = 3 + 4y and            3 + 4y = z, then we can apply the transitive property and make the connection that x = z. Further, triangles that are similar, are triangles whose only difference is size. Similar triangles will look like these have either been shrunk or are enlarged. Also, all the corresponding angles will be equal to one another. Thus, if the top angle of one triangle equals 45 degrees, then the top angles of all other similar triangles will also equal 45 degrees. If we compare the side lengths of one triangle to another, we will find that all the sides are proportional. In geometry, we can apply the transitive property to similar triangles to make connections. Let's see how this works. We start out with three triangles. We draw triangle A and then we draw a triangle B that is similar to triangle A. We then take triangle B and then we draw a triangle C that is similar to triangle B. So now we have three triangles: triangle A, triangle B, and triangle C. We know that triangle A is similar to triangle B and that triangle B is similar to triangle C. Applying the transitive property, it can be said that the triangle A is similar to triangle C. We can say this since the triangle A is similar to triangle B and triangle B is similar to triangle C, the only difference between the three triangles A, B and C is their size and they are all similar to one another. Thus, if A is similar to B and B is similar to C, the transitivity implies that A is similar to C.