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