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Common Core (Geometry)

Common Core for Mathematics

**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.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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

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 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.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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