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Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

Teresa D. Magnus

Chapter 2 Exploring Polygons and Other Figures on the Plane

In Section 1.1, we introduced two basic assumptions about area, Formula 1.1.2 and Principle 1.1.3. In Section 1.2, we added another principle Principle 1.2.1 as well as a long list of common vocabulary terms.
Before we continue, it will be useful to define what it means for two polygons to be congruent:

Definition 2.0.1.

Two polygons are congruent if they satisfy all of the following:
  • The polygons have the same number of sides.
  • Corresponding sides have the same length.
  • Corresponding angles have the same measure.
As an example, consider a pair of convex quadrilaterals \(ABCD\) and \(EFGH\text{.}\) We say \(ABCD\) is congruent to \(EFGH\text{,}\) written \(ABCD\cong EFGH\text{,}\) if the following are all true: \(AB=EF\text{,}\) \(BC=FG\text{,}\) \(CD=GH\text{,}\) \(DA=HE\text{,}\) \(m(\angle ABC)=m(\angle EFG)\text{,}\) \(m(\angle BCD)=m(\angle FGH)\text{,}\) \(m(\angle CDA)=m(\angle GHE)\text{,}\) and \(m(\angle DAB)=m(\angle HEF)\text{.}\) Note how the labeling of the quadrilaterals’ vertices identifies the pairs of corresponding sides and the pairs of corresponding angles.
Two congruent quadrilaterals with lengths and angle measures shown.
Figure 2.0.2. Congruent quadrilaterals \(ABCD\) and \(\cong EFGH\)
This leads us to another principle that we will accept as intuitively obvious:
In this chapter, we will show how these principles lead to the area formulas for other polygons. We will also use our knowledge of area to show why the Pythagorean Theorem holds. While you may already have these area formulas and the Pythagorean Theorem memorized, you will soon have first-hand knowledge of why they hold and how they fit with basic principles.
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