Examining Similar Triangles
Let us begin our exploration by defining similar figures. In order to prove that any two figures are similar, the measures of corresponding angles must be the same, but the measures of corresponding sides must be multiples (proportional) of each other.
For our example we are going to construct any triangle ABC:
In this example, let the lengths of the sides of triangle ABC be as follows:
m BA @ 5 cm
m BC @ 8 cm
m AC @ 4 cm
Let the measurements of the interior angles of triangle ABC be as follows:
m angle ABC @ 30 degrees
m angle BAC @ 120 degrees
m angle ACB @ 40 degrees
If we take the midpoints of this triangle and construct segments from the midpoints to the vertices, then we can copy these segments to construct a new triangle DEF, as noted below:
For the new triangle DEF, the lengths of the sides are as follows:
m FE @ 3 cm
m DE @ 6 cm
m DF @ 2 cm
The corresponding angles are congruent.
Therefore, these two triangles constructed from one triangle are similar, as the sides are in proportion and the angles are congruent.