Triangle Proportionality, and Similarity: Side-Side-Side, Side-Angle-Side, Angle-Angle, Triangle Midpoint Segment Theorem, and Proportionality in Parallel Lines Cut by Transversal.

Have you experience to be trying to solve a problem that has several triangles embedded in the figure, and you are required to find why some of them are similar, and in this way find the side lengths of those triangles in the figure? Sounds complicated?

This lesson takes the complexity out of it. You will learn the triangle similarity theorems with a lot of visual help to follow all the show. You will view in detail how the proportions are setup and solved. You will like it!

Lesson's Content

Lesson's Glossary

Angle-angle-angle (AAA) similarity
The angle-angle-angle (AAA) similarity test states that given two triangles that have corresponding angles that are congruent, then the triangles are similar. As we know the sum of the interior angles in a triangle is 180°, so if two corresponding are congruent, then the other ones should be as well.

Side-angle-side (SAS) similarity
The side-angle-side (SAS) similarity test states that given two triangles that have two pairs of sides that are proportional and the included angles are congruent, then the triangles should be similar.

Side-side-side (SSS) similarity
The side-side-side (SSS) similarity test states that for two triangles to be similar; all corresponding sides should be proportional.

Similar
Two polygons are similar polygons if corresponding angles have the same measure and corresponding sides are in proportion.

Similar triangles
Similar triangles are triangles which have the same shape but probably different size. Corresponding angles need to be congruent, and corresponding sides are in proportion.

Interactive Geometric Applets: Relevant Theorems.

An enlargement is when you multiply by a scale factor all the dimensions of a given

figure; whereas that a reduction requires that you divide by the scale factor. The resulting

figures are similar figures, because they have the same shape but different size.

This applet will enable you to visualize this relationship of similarity between two

triangles.

Parallel lines cut by transversals form proportional segments.

Verify that this is true by dragging point "E" in this applet.

Pay attention to the proportion see how it updates values as you drag the point.

The Basic Proportionality Theorem states that if you have a segment

connecting two sides of a triangle, and parallel to the third side; then

the segments formed are proportional.

Prove this theorem by dragging any point in the below figure.

Look at the proportion written below, it changes as you move the points.

The Triangle Midsegment Theorem states that if you have a segment joining two sides of the triangle by the midpoint,

then the third side is parallel to this segment, and measures twice the length of the midsegment.

Drag in this figure any of the vertices and verify these relationships.

Vocabulary Puzzle Interactive

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