Shape 

Copyright David & Cynthia Thomas, 2009

 

Chapter One: Summary

 

Chapter One began by asking four questions.  Those questions are …

1.      Given a particular set of segments and/or angles, is it always possible to construct a triangle?

2.      What necessary conditions must be met in order to construct and/or duplicate a particular triangle?  [Note: A complex procedure may include several necessary steps.]

3.      What sufficient conditions guarantee that a particular triangle may be constructed and/or duplicated? [Note: If a set of actions is sufficient, it must include all necessary steps.]

4.      Is it ever possible to construct more than one triangle from a given set of segments and/or angles?

 

The answers to those questions may now be summarized as follows:

1.      No.  The segments must satisfy the Triangle Inequality Theorem and the angles must have a sum of 180°.

2.      Every triangle has three segments and three angles (i.e., SSSAAA).  In order to construct and/or duplicate any triangle, you need to know at least three of these features.  Note: Not every combination of three features is sufficient.

3.      The following combinations of component features are sufficient to guarantee a unique triangle may be constructed and/or duplicated:

·        Side-Side-Side

·        Side-Angle-Side

·        Angle-Side-Angle

·        Angle-Angle-Side

4.      Yes.  The following combinations of component features may be used to create multiple triangles none of which are identical.

·        Side-Side-Angle

·        Angle-Angle-Angle