Two Column Proofs Involving Triangle Similarity and Proportionality.

How many times you have felt lost when trying to complete a two column proof? ...There are several statements in the given! which should be first?

In this lesson, you may very well forget about those headaches. We approach two column proofs involving triangle similarity in an innovative way. The proofs to study are presented showing you every single step, and how this is influenced by previous steps... until of the completion of the proof; after the example is over, you will be given the same proof with different labels in the figure, so that you may check your understanding of the solution process. You should really try 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.

Postulate
A statement assumed to be true without proof.

Proof
A sequence of justified conclusions used to prove the validity of an if-then statement.

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.

Theorem
A theorem in mathematics is a proven fact. A theorem about polygon must be true for every polygon; there can be no exceptions. An idea which works in several different cases is not enough.

Interactive Geometric Applets: Relevant Theorems.

Two parallel lines that are cut by a transversal have all acute angles

congruent, and all obtuse angles congruent. If you take one acute, and

one obtuse; their sum is 180°. In other words angle pairs like corresponding,

alternate interior, and alternate exterior angles are congruent, an consecutive interior

angles are supplementary.

You may drag point "B" in this applet to verify all these angle relationships.

Supplementary angles are two positive angles whose sum is 180°.

Drag point "C" within the shaded area to verify that you have

supplementary angles. Check the sum below the figure.

Congruent Supplements Theorem states that given two angles supplementary to the

same angle, then they are congruent between them.

Vertical Angles are a pair of nonadjacent angles formed by two straight lines like in the applet below.

Move point "C" right and left, and see how the vertical pairs remain congruent.

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