Polygons and their Properties
Introduction
In Chapter Two, we defined a polygon as follows:
Polygon: A polygon is a many sided closed figure comprised completely of line segments. Exactly two
line segments meet at each endpoint. Each mentioned endpoint is called a vertex and each mentioned
segment is called a side of the polygon.
In this chapter we would like to study polygons in detail and discover some of their properties and also
study triangles and quadrilaterals as two special classes of polygons.
4.1. Polygons
Polygons are all round us. They can be used to convey information:
Knowing the color and shape will allow you to recognize what
information will be presented before you can actually read the sign
while you are driving. The nature is full of polygons that convey its
sense of symmetry and design. We study polygons because they occur
in our world.
A convex polygon has the property that the line segment PQ generated
from any two arbitrary points P and Q inside the polygon lies entirely
inside the polygon. (Figure 4.1.2.a)
Figure 4.1.1
Q
B
P
A
(a)
Figure 4.1.2
(b)
In the case where we can find two points A and B inside of a polygon such that the line segment AB does
not lie entirely inside the polygon, the polygon is called nonconvex. (Figure 4.1.2.b)
From now on, by polygon we mean a convex polygon unless
otherwise stated.
A diagonal of a polygon is a line segment that joins two nonadjacent vertices.
In order to refer to a polygon with respect to its number of sides n,
we use the word n-gon. Therefore a triangle is a 3-gon and we do
not have a 2-gon in Euclidean geometry (We have this case in
spherical geometry!).
Figure 4.1.3
Theorem 4.1: Any n-gon can be divided into n 2 triangles.
To convince someone to accept this assertion, we should show the case for a general n. But n is a
variable and not a fixed value. What we can do is to study several specific n-gon cases such as heptagon
(7-gon) and octagon (8-gon) and then try to generalize the idea for all other cases.
A2
In general, we can say if we have an n-gon, then
we have n vertices that we can label them from A1
to An. The first two sides of A1A2 and A2A3 with
the diagonal A1A3 identify our first triangle. (We
used two sides to build one triangle—we lost one
extra side compared to the constructed triangles).
After that, each side, with the help of two adjacent
diagonals identify one new triangle (Here, we use
one side to build one triangle). But the last triangle
will employ the last two sides of A1An and AnAn1 to
be constructed (two sides for one triangle—we lost
another extra side here to gain a triangle). This
shows the number of constructed triangles are two
less than the number of sides!
A1
A3
An
An - 1
Figure 4.1.4
The following is an important theorem that gives us the ability to calculate the sum of angle measures of
any polygon:
Theorem 4.2: The sum of angle measures of an n-gon is (n  2)180.
To show that this theorem is valid we notice that from the previous theorem we can divide the n-gon to n
 2 triangles. In the next section we will show that the sum of angle measures of each triangle is 180.
Therefore, the sum of angle measures of an n-gon is (n  2)180.
The following table summarizes what we have studied related to the past two theorems.
48
Table 4.1.1
Number
Number of Diagonals
Number of Constructed
of Sides Emanated from a Vertex
Triangles
3
4
5
.
.
.
n
0
1
2
.
.
.
1
2
3
.
.
.
n3
n2
Sum of Angle Measures
180
2  180 = 360
3  180 = 540
.
.
.
(n  2) 180 = (n  2)180
Regular Polygons
A regular polygon is a polygon with all sides congruent and all angles congruent. We have already been
introduced to regular polygons such as equilateral triangles, squares, regular pentagon, and regular
hexagon.
We notice since all angles of a regular n-gon are congruent and since its sum of angle measures is (n 
2)180, we are able to find the degrees measure of each angle as follows:
Number of Degrees in one Angle = (n  2)180/n
Note: In some literatures the angle of a polygon is called “interior angle”.
Using the above formula, Table 4.1.2 presents information about some regular polygons.
Table 4.1.2
Regular Polygon
Equilateral Triangle
Square
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Number Number of Degrees
of Sides
in One Angle
3
60
90
4
108
5
120
6
128
4/7
7
135
8
140
9
144
10
The center of a regular polygon is a point inside of the polygon that is equidistance from all the vertices.
An angle with a vertex at the center of a regular polygon and with its sides passing through two adjacent
vertices of the polygon is called a central angle. We note that a regular n-gon contains n congruent
central angles. The sum of angle measures of all central angles is 360. Therefore we are able to
establish the following formula:
Number of Degrees in one Central Angle = 360/n
49
O
O
Figure 4.1.5. Central Angles
Rotational Symmetry and Regular Polygons
Suppose that you trace the design in Figure 4.1.6 by a pencil and then rotate the tracing design, which is
on the top of the original one, about its center, O. You will notice that in less than one full turn the
tracing design and the original figure will coincide. In fact, this will happen exactly three times. Such a
figure is said to have rotational symmetry. The point O is called the center of rotation. Since we could
coincide the tracing and original designs three times in one full turn, we say that the figure has three-fold
(3-fold) rotational symmetry.
All regular polygons have rotational symmetry.
The center of rotation is the center of the
polygon. A regular n-gon has n-fold rotational
symmetry.
O
Figure 4.1.6
Figure 4.1.7
50
Reflective Symmetry and Regular Polygons
Consider that you fold this page along the vertical dashed line in
Figure 4.1.8. Then you will notice that one half of the figure will
fit exactly over the other half. If you place a mirror along the
dashed line, you would observe that the mirror image of one half
of the design is exactly the same as the other half. We say that
such a design has reflective symmetry. The dashed line l is said
to be the line of reflection. Since the design has only one line of
reflection we say that it has one-fold (1-fold) reflective
symmetry.
All regular polygons have reflective symmetry. The lines of
reflections are concurrent at the center of the polygon. A regular
n-gon has n-fold reflective symmetry.
Figure 4.1.8
Figure 4.1.9
Exercise Set 4.1
1. Find the sum of angles of a heptagon, nonagon, 52-gon, and 100-gon.
2. Use only a compass and straightedge to construct a regular 3-gon, 6-gon, and 12-gon (dodecagon).
Repeat the problem by using the Geometer’s Sketchpad.
3. Find the angle and the central angle of the following regular polygons:
Figure 4.1.10
51
4. Find the measures of angles and central angles of a regular
(a) pentagon, (b) heptagon, (c) nonagon, (d) dodecagon, (e) and 60-gon.
5. For each of the designs in Figure 4.1.11 do as follows:
(a) If it has rotational symmetry, find its center of rotation and the number of its rotational
symmetry folds.
(b) If it has reflective symmetry, find its lines of reflection and the number of its reflective
symmetry folds.
Figure 4.1.11
6. Recall the theorem about the constructible regular polygons mentioned in Chapter Three that was
proved by Gauss. List all regular polygons with fewer than 50 sides that can be constructed with compass
and straightedge.
7. Using Sketchpad we are able to construct shapes with rotational symmetries. Take a look at Figure
4.1.12.a. It is a nonconvex polygon that has a right angle. Generate this polygon in the Geometer’s
Sketchpad (except for the right angle you don’t need to be accurate to produce it exactly). Now select O
and then in “Transform” menu click on “Mark Center”. Now select the entire figure and go back to
“Transform” but this time choose “Rotate”. A screen will appear and asks you about the angle of
rotation. Choose 90 and then click OK. You will see that if you repeat this procedure two more times
you have Figure 4.1.12.b, which has 4-fold rotational symmetry (and not reflective symmetry).
52
O
(a)
(b)
Figure 4.1.12
Now construct a noncovex polygon with a certain angle to create the following:
(a) A design with 8-fold rotational symmetry that does not have reflective symmetry.
(b) A design with 6-fold rotational symmetry that does not have reflective symmetry.
(c) A design with 5-fold rotational symmetry that does not have reflective symmetry.
8. Using Sketchpad we also are able to construct shapes with reflective symmetries. Consider Figure
4.1.13.a. It is a nonconvex polygon that has a right angle. Generate this polygon in the Geometer’s
Sketchpad. Now select the line segment AB and then in “Transform” menu click on “Mark Mirror”.
Then select your entire figure and go back to “Transform” and this time choose “Reflect”. You will see
that if you repeat this procedure by selecting BC and its mirror image as your new “Mirror” you will
obtain Figure 4.1.13.b, which has 2-fold reflective symmetry.
A
C
B
(a)
(b)
Figure 4.1.13
Now construct a noncovex polygon with a certain angle to create the following:
(a) A design with 4-fold reflective symmetry.
(b) A design with 3-fold reflective symmetry.
(c) A design with 6-fold reflective symmetry.
53
4.2. Triangles
In Chapter Two we defined triangles as follows:
Triangle ABC ( ABC): Union of three segments determined by three non-collinear points.
We are already familiar with several special triangles such as right triangle and equilateral triangle due to
our backgrounds in elementary and secondary school geometries. However, for the consistency of our
course of study we define all different types of triangles before we start.
Equilateral Triangle: A triangle with three congruent sides (Figure 4.2.1.a).
Isosceles Triangle: A triangle with (at least) two congruent sides (Figure 4.2.1.b).
From the above definitions it is obvious that any equilateral triangle is an isosceles triangle but not vice
versa.
Scalene Triangle: A triangle with no congruent sides (Figure 4.2.1.c).
Right Triangle: A triangle with a right angle (Figure 4.2.1.d).
Acute Triangle: A triangle with three acute angles (Figure 4.2.1.e).
Obtuse Triangle: A triangle with an obtuse angle (Figure 4.2.1.f).
(a)
(d)
(b)
(e)
(c)
(f)
Figure 4.2.1
Triangles are all around us. The ancient music scale of Pythagoras, the Pythagorean Diatonic Scale,
which is the origin of our today’s even-tempered tuning system of music, is based on the ratios of string
lengths involving the integers 1, 2, 3, and 4 which made up the tetraktys, a triangular array of 10 points
(Figure 4.2.2.a). A regular pentagon is divided into one type 1 golden triangle, an isosceles triangle of
sides in the ratio of   (the congruent sides are ), and two type 2 golden triangles in the ratio of  (the
congruent sides are 1). These triangles have been used in art and architecture over the centuries (Figure
54
4.2.2.b). The Vesica Piscis, a sacred architecture fish-shaped form is the region in common to two
intersecting circles of congruent radii whose centers lie on each other’s circumference, and inscribes two
attached congruent equilateral triangles (Figure 4.2.2.c-d).


(a)
1
(b)
(c)
(d)
Figure 4.2.2
There are several theorems that we are able to establish for properties of triangles. The first one that we
bring here is as follows:
Theorem 4.3: The sum of angles of any triangle is 180º.
Based on the theorems that have been presented in previous chapters we are able to prove this new one.
For this, consider the arbitrary triangle of  ABC (Figure 4.2.3.a). We construct the parallel line l to BC
that passes through A (Figure 4.2.3.b). Then we notice that  B   A1 because they are alternative
interior angles. The same is true for the pair of angles  C and  A2 (Figure 4.2.3.c). But m  A1 + m 
A + m  A2 = 180º. Therefore, the sum of angles  A, B, and  C is 180º.
A
A
l
l
A
2
1
B
B
(a)
C
C
C
B
(b)
(c)
Figure 4.2.3
55
In Chapter One, we mention that the five-axiom system of Euclid as is presented in his book Elements, is
not complete and we need more axioms for that purpose. The following is one of these axioms, which is
about triangles.
Axiom (SAS Postulate): Two triangles with side-angle-side correspondence are congruent.
Employing the above axiom and previous theorems, then, we are able to prove the following theorems:
Theorem 4.4 (ASA Theorem): Two triangles with angle-side-angle correspondence are congruent.
Theorem 4.5 (SSS Theorem): Two triangles with side-side-side correspondence are congruent.
Theorem 4.6 (SAA Theorem): Two triangles with side-angle-angle correspondence are congruent.
The following theorems are also used frequently in geometry. Among them is the most famous theorem
of the plane geometry, the Pythagorean Theorem, that we presented it in Chapter One.
Theorem 4.7: The opposite angles of the congruent sides in a triangle are congruent.
Theorem 4.8: The opposite sides of the congruent angles in a triangle are congruent.
Theorem 4.9 (Pythagorean Theorem): The sum of the squares of the two sides of a right triangle
equals the square of its hypotenuse.
To study triangles further, we need more definitions:
Median of a Triangle: A line segment from a vertex to the midpoint of the opposite side (Figure
4.2.4.a).
Altitude of a Triangle: A line segment from a vertex perpendicular to the line of opposite side (Figures
4.2.4.b and 4.2.4.c).
Angle Bisector of a Triangle: The angle bisector of an angle of a triangle (Figure 4.2.4.d).
(It may be indicated by a ray or a segment from the vertex of the bisected angle to a point on opposite
side.)
Perpendicular Bisector of a Triangle: The perpendicular bisector of a side of a triangle (Figure
4.2.4.e).
(a)
(b)
(c)
56
(d)
(e)
Figure 4.2.4
Exterior Angle of a Triangle: An angle that forms a linear pair with one of the angles of the triangle
(Figure 4.2.5.a).
Remote Interior Angle (with respect to an Exterior Angle): The two angles of the triangle which are
not adjacent to the exterior angle (Figure 4.2.5.b).
A
A
B
B
C
C
(a)
(b)
Figure 4.2.5
The following theorems present important properties of the above triangle components:
Theorem 4.10: The medians of a triangle are concurrent. (Figure 4.2.6.a)
Theorem 4.11: The lines containing the altitudes of a triangle are concurrent. (Figure 4.2.6.b)
Theorem 4.12: The perpendicular bisectors of a triangle are concurrent. (Figure 4.2.6.c)
Theorem 4.13: The angle bisectors of a triangle are concurrent. (Figure 4.2.6.d)
H
G
(a)
(b)
57
P
O
(c)
(d)
Figure 4.2.6
We further define more terms in relationship with triangles.
Centroid (G): The point of intersection of the medians in a triangle (Figure 4.2.6.a).
Orthocenter (H): The point of intersection of the lines containing the altitudes in a triangle (Figure
4.2.6.b).
Circumcenter (O): The point of intersection of the perpendicular bisectors (Figure 4.2.6.c).
Incenter (P): The point of intersection of the angle bisectors in a triangle (Figure 4.2.6.d).
The angle bisectors and the medians of a triangle always meet in the interior of the triangle. The lines
containing the altitudes and perpendicular bisectors can meet in the exterior or on the triangle itself.
H
H
(a)
(b)
O
O
(c)
(d)
Figure 4.2.7. (a-b) Altitudes that meet outside or on the triangle, (c-d) Perpendicular bisectors that meet
outside or on the triangle.
58
Some Properties of Triangles
The circumcenter O is the center of a circle called the Circumcircle that contains all three vertices of a
triangle.
The incenter P is the center of a circle called the Incircle in a triangle that meets each side of the triangle
in only one point that is not a vertex (the circle is tangent to each side).
P
O
Figure 4.2.8
We also are able to establish the following theorem, which exhibits a relationship among the centroid,
orthocenter, and circumcenter of a triangle.
Theorem 4.14: The centroid, orthocenter, and circumcenter of a triangle are collinear and G lies 2/3 of
the distance from H to O.
The line that contains these three points is called the Euler Line, in memory of the Swiss mathematician
Leonhard Euler (1707—1763 A.D.) who discovered this relationship.
H
G
O
Figure 4.2.9
59
Exercise Set 4.2
1. Which of the following statements are always true, sometimes true, or never true.
a. A right triangle is an equilateral triangle.
b. A right triangle is an isosceles triangle.
c. An isosceles triangle is an equilateral triangle.
d. An equilateral triangle is an isosceles triangle.
e. An equilateral triangle is an acute triangle.
f. An acute triangle is an equilateral triangle.
g. A scalene triangle is a right triangle.
h. An equilateral triangle is a scalene triangle.
i. An isosceles triangle is a right triangle.
A Venn Diagram is a diagram that enables us to see the relationships among objects. For example, the
following Venn Diagram shows the relationship between isosceles triangles I, and equilateral triangles E.
It presents that elements in smaller circle are members of the larger circle; however, we may locate
members of the larger circle I which are not in E. This means every equilateral triangle is an isosceles
triangle but not vice versa.
I
E
Figure 4.2.10
2. Draw a Venn Diagram that shows the relationships among right triangles R, equilateral triangles E,
scalene triangles S, and the set of all triangles U.
3. Draw a Venn Diagram that shows the relationships among right triangles R, acute triangles A, obtuse
triangles O, equilateral triangles E, and the set of all triangles U.
4. Draw a Venn Diagram that shows the relationships among isosceles triangles I, acute triangles A,
obtuse triangles O, equilateral triangles E, and the set of all triangles U.
5. Use Sketchpad to construct an equilateral triangle, a right triangle, an acute scalene triangle, and an
obtuse triangle. Then perform the following for each of the constructed triangles.
a.
b.
c.
d.
Find the circumcircle of the triangle.
Find the incircle about the triangle.
Find the Euler line.
Check if G lies 2/3 of the distance from H to O.
60
6. Using the Geometer’s Sketchpad construct a triangle and find its centroid G and then do the following:
(a) Use
and drag the vertices around. Is it possible for G to lie on a vertex of the triangle? If
so, explain under what circumstances? If not, explain why not.
(b) Is it possible for G to lie on a side of the triangle? If so, explain under what circumstances? If
not, explain why not.
(c) Is it possible for G to lie outside of the triangle? If so, explain under what circumstances? If
not, explain why not.
7. Using the Geometer’s Sketchpad construct a triangle and find its orthocenter H and then do the
following:
(a) Use
and drag the vertices around. Is it possible for H to lie on a vertex of the triangle? If
so, explain under what circumstances? If not, explain why not.
(b) Is it possible for H to lie on a side of the triangle? If so, explain under what circumstances? If
not, explain why not.
(c) Is it possible for H to lie outside of the triangle? If so, explain under what circumstances? If
not, explain why not.
8. Using the Geometer’s Sketchpad construct a triangle and find its circumcenter O, and then construct
its circumcircle. Now do the following:
(a) Use
and drag the vertices around. Is it possible for O to lie on a vertex of the triangle? If
so, explain under what circumstances? If not, explain why not.
(b) Is it possible for O to lie on a side of the triangle? If so, explain under what circumstances? If
not, explain why not.
(c) Is it possible for O to lie outside of the triangle? If so, explain under what circumstances? If
not, explain why not.
Note: In order to construct the circumcircle you need to use
, start from circumcenter O and then drag
to one of the vertices. Then the generated circle will pass through the other two vertices.
9. Using the Geometer’s Sketchpad construct a triangle and find its incenter P, and then construct its
incircle. Now do the following:
(a) Use
and drag the vertices around. Is it possible for P to lie on a vertex of the triangle? If
so, explain under what circumstances? If not, explain why not.
(b) Is it possible for P to lie on a side of the triangle? If so, explain under what circumstances? If
not, explain why not.
(c) Is it possible for P to lie outside of the triangle? If so, explain under what circumstances? If
not, explain why not.
Note: In order to construct the incircle you need to construct a perpendicular line from the incenter P to
one of the three sides. Then construct the point of intersection of the perpendicular line and this side.
61
Next use
, start from P and then drag
to this constructed intersection. Then the generated circle
will meet the other two sides each in one point. You may construct these two points.
A
10. Draw  ABC. Select an arbitrary point on
AB and call it D. Draw a parallel line to BC that
passes through D and meets AC at E. Find AD /
AB , AE / AC , and DE / BC . What can you say
E
D
about these values? What can you say about the
corresponding angles in two triangles  ADE and 
ABC.
B
C
Figure 4.2.11
a
11. The following figure presents another
method of proving the Pythagorean Theorem
which was produced by the twentieth
president of the United States, James A.
Garfield (1831-1881). It is based on the
formula for the area of a trapezoid A = ½
(sum of bases)(altitude). We will study this
formula in Chapter Seven. Use the following
figure to write a short essay to convince
b
someone that the Pythagorean Theorem is true.
For this, you need to present the area of the
trapezoid one time based on the mentioned
formula, and the other time based on the sum of
areas of three triangles, and compare.
a
c
c
b
Figure 4.2.12
4.3. Quadrilaterals
The following definition appeared in Chapter Two:
Quadrilateral ABCD: Union of four segments in a plane determined by four points, no three of which
are collinear. The segments intersect only at their endpoints.
Because of the extensive use of quadrilaterals in our lives we study them as a special class of polygons
and based on their properties we divide them into several groups:
62
Trapezoid: A quadrilateral with only one pair of parallel sides (Figure 4.3.1.a).
Parallelogram: A quadrilateral with two pairs of parallel lines (Figure 4.3.1.b).
Kite: A quadrilateral with two pairs of adjacent congruent sides (Figure 4.3.1.c).
Rhombus: A quadrilateral with all congruent sides (Figure 4.3.1.d).
Rectangle: A quadrilateral with all congruent angles (Figure 4.3.1.e).
Square: A regular quadrilateral, all sides are congruent, all angles are congruent (Figure 4.3.1.f).
(a)
(b)
(d)
(c)
(e)
(f)
Figure 4.3.1. A trapezoid, a parallelogram, a kite, a rhombus, a rectangle, and a square.
Based on the previous theorems and properties associated with the above definitions we are able to
establish the following:
Theorem 4.15: All angles of a rectangle are right angles, and therefore a rectangle is a parallelogram.
Theorem 4.16: A rhombus is a parallelogram.
We notice that we have more relationships among these quadrilaterals. For example, each square is a
rectangle, and also it is a rhombus as well. Therefore, we are able to present the following Venn
Diagram.
Rectangles
Rhombi
Squares
Figure 4.3.2
63
The following are more theorems about quadrilaterals:
Theorem 4.17: The opposite sides of a parallelogram are congruent.
Theorem 4.18: The opposite angles of a parallelogram are congruent.
Theorem 4.19: The diagonals of a parallelogram bisect each other.
Theorem 4.20: The diagonals of a rectangle are congruent.
Theorem 4.21: The diagonals of a rhombus are perpendicular bisectors of each other.
In Chapter Three, we studied a special class of rectangles, Golden rectangles. Here is another group of
rectangles that is derived from the square. Rectangles in this group exhibit a sort of harmony and
proportionality among themselves and because of that have been used in art and architecture in different
eras. This group is identified as the Dynamic Rectangles. They are generated from a unit square (a
square with unit sides) using compass and straightedge as follows: We construct the unit square ABCD
and then we use the measure of its diagonal to construct the rectangle AEFD. The length of this rectangle
is 2 (why?). The next rectangle will be constructed using the diagonal of the rectangle AEFD. The new
rectangle AGHD has a length equal to 3 (why?). With the same procedure we can construct rectangles
with unit width and with length equal to n for any positive integer n and this construction can be carried
on indefinitely.
D
C
F
H
...
A
B
E
G
Figure 4.3.3
Exercise Set 4.3
1. Draw a Venn Diagram to present the relationship among kites, rhombi, and squares.
2. Draw a Venn Diagram to present the relationship among parallelograms, rectangles, and squares.
3. Write a short essay to convince someone that Theorem 4.15 is true.
4. Write a short essay to convince someone that Theorem 4.16 is true.
64
5. In Figure 4.3.4, locate the presented quadrilaterals in their correct locations.
All Sides
Congruent
All Angles
Congruent
Figure 4.3.4
6. Using the Sketchpad, first construct a square, and then use this square as the unit square to construct
the first five Dynamic rectangles.
7. Write a short essay to convince someone that Theorem 4.19 is true.
8. Write a short essay to convince someone that Theorem 4.20 is true.
9. What can you say about each of the following statements? Use one of the expressions true, sometimes
true, never true.
a. A rhombus is a rectangle.
b. A rhombus is a kite.
c. A rhombus is a square.
d. A square is a rectangle.
e. A rectangle is a square.
f. A rectangle is a kite.
g. A square is a rhombus.
h. A rectangle is a parallelogram.
i. A parallelogram is a trapezoid.
j. A square is a parallelogram.
k. A trapezoid is a kite.
10. Let ABCD be an arbitrary parallelogram. Draw the diagonal AC. How can you convince someone
that two triangles  ABC and  ACD are congruent?
The Tangram puzzle consists of seven pieces that all together make a square. It includes 5 triangles, one
parallelogram, and one square. The origin of the puzzle and its name are open to speculation. However,
it is widely believed that this puzzle originated in China, where people enjoyed it in the early 1800’s.
65
Figure 4.3.5
11. Using a Tangram puzzle construct the following:
a.
b.
c.
d.
A square using seven pieces.
A square using five pieces.
A square using four pieces.
A square using two pieces.
12. Using a Tangram puzzle construct the following:
a.
b.
c.
d.
e.
f.
A parallelogram using seven pieces.
A parallelogram using six pieces.
A parallelogram using five pieces.
A parallelogram using four pieces.
A parallelogram using three pieces.
A parallelogram using two pieces.
13. Using a Tangram puzzle construct the following shapes:
(a)
(b)
Figure 4.3.6
66
(c)