InterMath Dictionary

Similarity
 Angle-Angle (AA) Similarity Side-Side-Side (SSS) Similarity Congruent Side-Angle-Side (SAS) Similarity Similar Figures Corresponding Angles Similar Triangles

Similarity
 Congruent - Two angles or two sides are congruent if they have the same measure. Corresponding Angles - Angles that have the same relative positions in geometric figures. Corresponding Sides - Sides that have the same relative positions in geometric figures. Similar Figures - Figures that have the same shape but not always the same size.

Corresponding parts in a figure mean that the parts (angles or sides) are in the same relative position in each of the figures. For example, in the figure below there are two triangles: a larger triangle, ABC, and a smaller triangle, ADE. Angles ADE and ABC are corresponding because they both lie in the same relative position in the two triangles. Angles ABC and AED are not corresponding since they do not lie in the same relative position in the two triangles. For the same reason, sides AE and AC are corresponding while sides BC and AD are not.

Similar figures have the same shape, but not always the same size. Similar figures and congruent figures both have the same shape. The difference between similar figures and congruent figures is that congruent figures also have the same size. So, corresponding sides and corresponding angles of congruent figures have the same measures. The measures of corresponding angles in similar figures are the same, but the measures of corresponding sides in similar figures are multiples of each other (i.e. they are proportional). Therefore, in order to prove that any two figures are similar, one must show that these two conditions hold for all corresponding angles and all corresponding sides between the two figures.

For example, the two parallelograms below are similar figures. Notice that they have the same shape, but obviously not the same size. All four of their corresponding angles are congruent, and all four of their corresponding sides are proportional (here, in a ratio of 1 to 2, or 0.500).

Triangle Similarity

 Angle-Angle (AA) Similarity - If two angles of one triangle are congruent to two corresponding angles in another triangle, then the triangles are similar. Side-Angle-Side (SAS) Similarity - If the lengths of two sides of a triangle are multiples of two corresponding sides of another triangle (i.e. proportional), and the included angles are congruent, then the triangles are similar. Side-Side-Side (SSS) Similarity - If the lengths of all three of the corresponding sides of two triangles are multiples of each other (i.e. proportional), then the triangles are similar.

As the simplest type of polygon, triangles are unique in that there are certain properties that can be used to verify that two triangles are similar. Instead of showing that all corresponding angles are congruent and all corresponding sides are in proportion, one can simply use one of the following three properties. You will notice that these properties seem a little too familiar (see also Congruent Triangles)!

Angle-Angle (AA)

According to the AA similarity property, to show two triangles are similar, it is sufficient to verify that two angles of one triangle are congruent to the corresponding angles of another triangle. In other words, this tells you that the shape of a triangle is completely determined by the measures of its angles.

Side-Angle-Side (SAS)

According to the SAS similarity property, to show two triangles are similar, it is sufficient to verify that the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle, and the included angles are congruent.

Side-Side-Side (SSS)

According to the SSS similarity property, to show two triangles are similar, it is sufficient to verify that the measures of the three sides of one triangle are proprotional to the corresponding measures of the sides of another triangle.

Check for Understanding

 Determine which similarity property applies in each of the java applets. Drag around any of the red points to test a conjecture. Choose each answer from the menus a) Sorry, this page requires a Java-compatible web browser. Choose one AA similarity SAS similarity SSS similarity not enough information b) Sorry, this page requires a Java-compatible web browser. Choose one AA similarity SAS similarity SSS similarity not enough information c) Sorry, this page requires a Java-compatible web browser. Choose one AA similarity SAS similarity SSS similarity not enough information d) Sorry, this page requires a Java-compatible web browser. Choose one AA similarity SAS similarity SSS similarity not enough information

External Links to Extend Basic Understanding

Harcourt's Animated Math Glossary: Read another definition of similarity.

Similar Triangles at Math.com: See a lesson about similar triangles and then test yourself with the included questions.

Related Investigations on InterMath to Challenge Teachers

Similar Triangles: Determine proportions comparing the side lengths of similar triangles.
Side Splitter
: Find a relationship between sides in two triangles.
Altitude to the Hypotenuse: Investigate the relationship between sides of right triangles when an altitude is drawn to the hypotenuse of a right triangle.
Fractal Iterations
: Explore the idea of fractals.
Double the fun
: Double the dimensions of quadrilaterals.

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