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 . 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.
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§ 2. The Triangle
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The proof that the three angles of a triangle sum to is straightforward in Euclidean geometry. We start with a a triangle . See Figure 1.
Proof:

Construct .

Construct line DE parallel to side AC through point B; .

because they make a straight line.

because they are opposite interior angles formed by the transversal intersecting parallel lines and .

because they are opposite interior angles formed by the transversal intersecting parallel lines and .

Therefore .
QED
(Note that this proof fails in nonEuclidean 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.
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 , 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.
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§ 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.
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
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.
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§ 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.
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
We now have a formula that applies to all quadrilaterals just as we had a formula that applies to all triangles!
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§ 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.
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
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.
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§ 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 vertices. A polygon is a convex polygon if the angle measure of any interior angle is less than a straight line. See Figure 10.
In other word, the angle may be:

acute, meaning between and .

right, meaning exactly .

obtuse, meaning between and .
A polygon is not convex if it is concave at one or more vertices as shown in Figure 11.
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 and .
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§ 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 vertices, we obviously have one triangle. See Section 2.

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

For a convex polygon with vertices, we have triangles.
The sum of the interior angles of any polygon will be the same as the sum of the interior angles of its triangles. Therefore
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§ 8. Formula for a Regular Polygon
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With a regular polygon, its sides all have the same length, and its angles all have the same angle measure. See Figure 12.
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§ 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.
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
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.
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§ 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.
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
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.
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§ 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?
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§ 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
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