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Two figures are similar when there is a sequence of similarity transformations that map one figure to the other.
When two triangles are similar, corresponding angles are congruent and corresponding sides are proportional.
There are some “shortcuts” that make it easier to prove that two triangles are similar.
Recall that we have defined similarity as the characteristic of one geometrical object to be an image of another after transformation of scaling and, possibly, some congruent transformation (translation, rotation and symmetry relative to an axis). Applied to triangles, we see, first of all, that an image of a triangle after transformation of scaling is a triangle (since straight lines are transformed into straight lines). We also observe that similar triangles have corresponding angles congruent (since scaling (dilation) preserves angles) and corresponding sides proportional with the same coefficient of proportionality equal to a scaling factor (since transformation of scaling changes the lengths of all segments by the same scaling factor). All in all, scaling transforms a triangle into another triangle, similar to original (by definition of similarity), with correspondingly congruent angles and correspondingly proportional sides.
The property of triangles to have congruent angles and proportional sides is, actually, equivalent to their similarity. In fact, three much shorter statements are true, each one, obviously, necessary and, as we are going to prove, sufficient conditions for similarity of triangles.
Theorem 1. If two triangles have two pairs of angles correspondingly congruent to each other, then they are similar.
Theorem 2. If two triangles have one pairs of congruent angles and sides, forming these angles, are proportional, then they are similar.
Theorem 3. If three sides of one triangle are correspondingly proportional to three sides of another triangle, then these triangles are similar.
This video explains how to determine if two triangles are similar using AA similarity.
This video explains how to determine if two triangles are similar using SSS and SAS.
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