Examples, solutions, videos, and lessons to help High School students learn when given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Common Core: HSG-SRT.A.2
Similarity and Transformations
Use transformations to prove similarity.
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Geometry2D - Similarity
we can define similarity of two geometrical objects on a plane as
possibility to transform one into another using dilation (scaling),
as defined above, optionally combined with congruent transformations
of parallel shift (translation), rotation and symmetry (reflection).
Geometry2D - Similarity - Triangles
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
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
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
Theorem 3. If three sides of one triangle are correspondingly
proportional to three sides of another triangle, then these
triangles are similar.
Geometry - Similarity - Circles
Theorem 1 Scaling of a circle results in a circle. The radii of
these circles are related by a scaling factor.
Theorem 2 Any two circles are similar.
Theorem 3 For any two circles, positioned outside of each other,
there is a transformation of scaling that transforms one into the
Similarity of Circles
How to prove all circles are similar?
Quadrilateral similarity by showing congruent angles
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