Properties of Similar Triangles:
If , then the following holds true:
Corresponding angles are congruent (same measure)
Corresponding sides are all in the same proportion
Axioms of Similarity of Triangles:
1. SAS (Side Angle Side) Postulate: If two triangles have a pair of corresponding angles equal and the sides including them are proportional, then the two triangles are similar. Or we can also say “Triangles are similar if two sides in one triangle are in the same proportion to the corresponding sides in the other, and the included angles are equal.”
2. AAA (Angle Angle Angle) Postulate. It is also called AA postulate. If two angles are equal, then the third one is equal too. In our discussion, we will use AAA as our nomenclature so that there is no confusion. Or we can say “Triangles are similar if the measure of all three interior angles in one triangle are the same as the corresponding angles in the other.”
and (also implies
3. SSS (Side Side Side) Postulate: If in two triangles, the corresponding sides are proportional, then the two triangles are similar.
Theorem of Proportionality:
1. If a line is drawn parallel to any side of the triangle, then the line divides the other two sides of the triangle proportionally.
Conversely, If a line divides the two sides of a triangle proportionally, the line is parallel to the third side.
2. Also because the two triangles are similar i.e.
Relation between the areas of two similar triangles:
- A median divides the triangle into two triangles of the same area.
i.e. Area of
- If multiple triangles have a common vertex (as shown in the diagram) and their bases are along a straight line, then the following holds true:
Maps and Models:
The concept of the similarity applies to the models that we build. The models are similar to the real object just that they are small (to be manageable). In case of models:
For a given scale factor :
- If the transformation is an enlargement
- If the transformation is a reduction
- If the transformation is an identity transformation
- Each side of the resulting figure (model) is times the corresponding side of the given figure (object)
- The area of the resulting figure (model) is times the area of the given figure (object)
- In case of solids, the volume of the resulting figure (model) is times the volume of the given object