November 24, 2017

Polygon Angles

Polygon Angles in Euclidean Geometry
Robert F Szalapski, PhD

§ 1. Introduction

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A very important result in Euclidean Geometry, familiar to most any student, is that the angles of a triangle sum to 180^{0}. Armed with this knowledge we can discover the sum of the angle measures for other polygons with special results for the regular polygons. This is a very important topic for many areas of mathematics, science and engineering, but the results are pretty cool in their own right.

In this article I will explain briefly how to obtain the angle measures of polygons. In Section 12 I will link to the Geometer’s Sketchpad and Smart Notebook files that I developed with a colleague at Nazareth College. These items will be available for free download on another page. The figures on this page are obtained from screen shots of Geometer’s Sketchpad.

§ 2. The Triangle

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The proof that the three angles of a triangle sum to 180^{0} is straightforward in Euclidean geometry. We start with a a triangle \triangle ABC. See Figure 1.

Figure 1.

In Euclidean geometry we may prove that the angles of a triangle sum to 180^{0} by constructing a line parallel to one side and through the vertex opposite.

Proof:

  1. Construct \triangle ABC.

  2. Construct line DE parallel to side AC through point B; \stackrel{\leftrightarrow}{DE}\parallel\stackrel{\leftrightarrow}{AC}.

  3. \angle ABD+\angle ABC+\angle EBC=180^{0} because they make a straight line.

  4. \angle ABD=\angle CAB because they are opposite interior angles formed by the transversal \stackrel{\leftrightarrow}{AB} intersecting parallel lines \stackrel{\leftrightarrow}{DE} and \stackrel{\leftrightarrow}{AC}.

  5. \angle EBC=\angle BCA because they are opposite interior angles formed by the transversal \stackrel{\leftrightarrow}{BC} intersecting parallel lines \stackrel{\leftrightarrow}{DE} and \stackrel{\leftrightarrow}{AC}.

  6. Therefore \angle CAB+\angle ABC+\angle CBA=180^{0}.

QED
(Note that this proof fails in non-Euclidean geometry.)

A triangle was constructed in Geometer’s Sketchpad along with tables to display the angle and length measures. Screen shots may be viewed in Figure 2 and Figure 3, respectively.

Figure 2.

A screen shot of a triangle construction from Geometer’s Sketchpad. In the Sketchpad version the vertices may be manipulated changing the angle and length measures.

Figure 3.

A screen shot of a table construction from Geometer’s Sketchpad. In the Sketchpad version the table updates automatically as the vertices of the triangle are manipulated.

When working in Geometer’s Sketchpad the user may use the mouse to move the verticies of the triangle, and the angle and length measures in the tables will update automatically. The student will notice that the sum of the angles never deviates from 180.00^{0}, and this is an excellent visual illustration of what we proved above. The visual illustration augments the proof, but it does not replace it. We cannot hope to test every case as there are infinitely many possibilities. Furthermore, results are displayed to only two decimal places, and we do not know without the proof whether or not there could be a tiny deviation requiring more decimal places.

§ 3. The Equilateral Triangle

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Screen shots of an equilateral triangle and the tables with its side and angle measures appear in Figure 4 and Figure 5, respectively.

Figure 4.

A screen shot of an equilateral-triangle construction from Geometer’s Sketchpad.

Figure 5.

A screen shot of a table construction from Geometer’s Sketchpad. In the Sketchpad version the table updates automatically as the vertices of the equilateral triangle are manipulated.

As one manipulates a vertex of the triangle, the side lengths remain equal. Additionally all of the vertex measures remain equal with a value of

\displaystyle\frac{180^{0}}{3}=60^{0}\;.

Note that, within the Geometer’s Sketchpad file, the construction of the equilateral triangle may be exposed by selecting Show All Hidden from the Display menu.

§ 4. The Quadrilateral

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A quadrilateral was constructed in Geometer’s Sketchpad along with tables to display the angle and length measures. Screen shots may be viewed in Figure 6 and Figure 7, respectively.

Figure 6.

A screen shot of a quadrilateral construction from Geometer’s Sketchpad. In the Sketchpad version the vertices may be manipulated changing the angle and length measures.

Figure 7.

A screen shot of a table construction from Geometer’s Sketchpad. In the Sketchpad version the table updates automatically as the vertices of the quadrilateral are manipulated.

Notice how the quadrilateral may be subdivided into two triangles. Notice how the sum of the four interior angles of the quadrilateral is equal to the sum of the six interior angles of the pair of triangles. For this reason we see that, as the quadrilateral is reshaped and rescaled, the sum of the interior angles totals

\displaystyle 2\times 180^{0}=360^{0}\;.

We now have a formula that applies to all quadrilaterals just as we had a formula that applies to all triangles!

§ 5. The Square

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Screen shots of a square and the tables with its side and angle measures appear in Figure 8 and Figure 9, respectively.

Figure 8.

A screen shot of a square construction from Geometer’s Sketchpad.

Figure 9.

A screen shot of a table construction from Geometer’s Sketchpad. In the Sketchpad version the table updates automatically as the vertices of the square are manipulated.

As one manipulates a vertex of the square, the side lengths remain equal. Additionally all of the vertex measures remain equal with a value of

\displaystyle\frac{360^{0}}{4}=90^{0}\;.

Of course we already knew this, but to demonstrate it so clearly should give us confidence!
Note that, within the Geometer’s Sketchpad file, the construction of the equilateral triangle may be exposed by selecting Show All Hidden from the Display menu.

§ 6. A Convex Polygon

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More examples will be shown below, but let’s jump to the conclusion first. Consider a convex polygon with N vertices. A polygon is a convex polygon if the angle measure of any interior angle is less than a straight line. See Figure 10.

Figure 10.

Example of a convex polygon.

In other word, the angle may be:

  1. acute, meaning between 0^{0} and 90^{0}.

  2. right, meaning exactly 90^{0}.

  3. obtuse, meaning between 90^{0} and 180^{0}.

A polygon is not convex if it is concave at one or more vertices as shown in Figure 11.

Figure 11.

Example of a concave polygon.

A concave vertex occurs at an interior angle that is greater than a straight line. In other words, the vertex angle is a reflex angle meaning between 180^{0} and 360^{0}.

§ 7. Formula for a Convex Polygon

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In this section we jump to the formula for the sum of the interior angle measures of a convex polygon. More examples will be presented below for further illustration.

  • For a triangle, which has N=3 vertices, we obviously have one triangle. See Section 2.

  • Moving to a convex quadrilateral with N=4 vertices, we have two triangles. See Section 4.

  • For a convex polygon with N vertices, we have N-2 triangles.

The sum of the N interior angles of any polygon will be the same as the sum of the interior angles of its N-2 triangles. Therefore

\displaystyle\mathrm{sum\; of\;}N\mathrm{\; interior\; angles}=(N-2)\times 180^{0}

This agrees with the results of Section 2 and Section 4.

§ 8. Formula for a Regular Polygon

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With a regular polygon, its N sides all have the same length, and its N angles all have the same angle measure. See Figure 12.

Figure 12.

Examples of a regular polygons where all side measures are equal and all angle measures are equal.

The length of a side is not bounded. However, the measure of an interior angle may be calculated by dividing the total angle measure from Section 7 by the number of vertices:

\displaystyle\mathrm{one\; angle}=\frac{(N-2)\times 180^{0}}{N}

This agrees with the results of Section 3 and Section 5.

§ 9. The Pentagon

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A convex pentagon was constructed in Geometer’s Sketchpad along with tables to display the angle and length measures. Screen shots may be viewed in Figure 13 and Figure 14, respectively.

Figure 13.

A screen shot of a convex pentagon construction from Geometer’s Sketchpad. In the Sketchpad version the vertices may be manipulated changing the angle and length measures.

Figure 14.

A screen shot of a table construction from Geometer’s Sketchpad. In the Sketchpad version the table updates automatically as the vertices of the pentagon are manipulated.

Notice how the pentagon may be subdivided into three triangles. Notice how the sum of the five interior angles of the pentagon are equal to the sum of the interior angles of those triangles. For this reason we see that, as the pentagon is reshaped and rescaled, the sum of the interior angles totals

\displaystyle 3\times 180^{0}=540^{0}\;.

This agrees with the general formula presented in Section 7. Note that the pentagon must remain convex for this formula to apply. Some modification would be required for concave vertices.

§ 10. The Regular Pentagon

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Screen shots of a regular pentagon and the tables with its side and angle measures appear in Figure 15 and Figure 16, respectively.

Figure 15.

A screen shot of a regular pentagon construction from Geometer’s Sketchpad.

Figure 16.

A screen shot of a table construction from Geometer’s Sketchpad. In the Sketchpad version the table updates automatically as the vertices of the regular pentagon are manipulated.

As one manipulates a vertex of the regular pentagon, the side lengths remain equal. Additionally all of the vertex measures remain equal with a value of

\displaystyle\frac{540^{0}}{5}=108^{0}\;.

This agrees with the results of Section 8.
Note that, within the Geometer’s Sketchpad file, the construction of the regular pentagon may be exposed by selecting Show All Hidden from the Display menu.

§ 11. Extending the Results

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Think about how these results might be extended. When subdividing polygons into triangles, we used diagonals that do not intersect. How would the analysis change we allowed for diagonals to intersect? How would the analysis change if we allowed for concave vertices?

§ 12. Getting the Files

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The files are available on the Polygon Angles page in the area set aside for teachers. Smart Notebook, Geometers’s Sketchpad and PDF files are available for free download. Suggestions for how to use the files are included. The Smart Notebook file adds scaffolding, and overall the lessons are very inquiry based with many checkpoints to test student understanding. A dynamic table for the SmartBoard is shown in Figure

Figure 17.

An interactive table for the SmartBoard for exploring the topic of polygon angles.

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