Related Topics:
Common
Core (Geometry)
Common Core
for Mathematics
Videos, examples, solutions, and lessons to help High School students learn how to:
• Use the properties of similarity transformations to establish the
AA criterion for two triangles to be similar.
• Use the properties of similarity transformations to establish the AA criterion for two
triangles to be similar.
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.
AA Similarity Criterion: If two angles of one
triangle are congruent to two angles of another triangle, then the
triangles are similar.
SAS Similarity Criterion: If two sides of one
triangle are proportional to two sides of another triangle and
their included angles are congruent, then the triangles are
similar.
SSS Similarity Criterion: If three sides of one
triangle are proportional to three sides of another triangle, then
the triangles are similar.
Common Core: HSG-SRT.A.3
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
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.
Similar Triangles Using Angle-Angle
This video explains how to determine if two triangles are similar
using AA similarity.
Similar Triangles Using Side-Side-Side and Side-Angle-Side
This video explains how to determine if two triangles are similar
using SSS and SAS.
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