To understand the concept of similarity of triangles, one must think of two different concepts: shape and scale. In particular, similar triangles are triangles that have the exact same shape but may or may not have the scale. In other words one of the triangles is an enlarged version of the other triangle. For a triangle, the shape is determined by its angles, so the statement that two triangles have the same shape simply means that their corresponding angles are equal.
If two triangles have two corresponding pairs of angles with the same measure then they are similar. Since two angles of equal measure implies equality of the third.
For 
to be similar to 
, the following conditions must hold true:
to be similar to , the following conditions must hold true:
Corresponding Angles and Sides
Corresponding angles are a pair of matching angles that are in the same spot in a triangle. The corresponding sides are the sides opposite the angle. For example, in the above example, 
and the corresponding sides for the two angles are 
and 
. Similarly 
and the corresponding sides are 
and 
. When the triangles are similar the corresponding sides are in proportion i.e.:
and the corresponding sides for the two angles are and . Similarly and the corresponding sides are and . When the triangles are similar the corresponding sides are in proportion i.e.:
The triangles shown below are similar but may not appear as similar because one of the triangles has been rotated around a vertex. Therefore always check to see if the corresponding angles are equal to conclude if two triangles are similar or not.
Similarity and Congruency
Similar triangles do not necessarily have congruent sides. As we explained before, if triangles are similar one of the triangles could be an enlarged version of the other triangle.
Example:
What is the value of 
based on the figure shown above?
based on the figure shown above?
A) 2
B) 3
C) 4
D) 5
E) 6
Solution:
Step 1: From the figure we know that 
. We also know that 
and 
are right angles. Since two of the angles are identical, we can say that 
and 
are similar.
. We also know that and are right angles. Since two of the angles are identical, we can say that and are similar.
Step 2: When triangles are similar, the sides corresponding to equal angles are proportional. The angle opposite side is 
. The corresponding side for is 10. Similarly the sides opposite the right angle are 6 and 15 respectively. The following figure shows the corresponding sides of the two triangles in the same color.
is . The corresponding side for is 10. Similarly the sides opposite the right angle are 6 and 15 respectively. The following figure shows the corresponding sides of the two triangles in the same color.
Since the corresponding sides of similar triangles are in proportion, we get:
After cross-multiplying and simplification we get:
C is the right answer.
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