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TOPIC 11
Triangles and polygons
We can proceed directly from the results we learned about parallel lines and their associated
angles to the ideas concerning triangles.
An essential question is a question whose answer is not obvious, but it is a vitally important
question that we continue to ponder.
Essential question: Why are triangles so important in the study of
geometry?
We will spend more time studying triangles than any other single topic. Why?
Let’s review the definitions:
1.
What is a triangle?
2.
What is an isosceles triangle?
3.
What is an equilateral triangle?
4.
What is a scalene triangle?
5.
What is a scalene triangle?
6.
What is an acute triangle?
7.
What is an obtuse triangle?
8.
What is a right triangle?
Topic 11 (Triangles and Polygons)
page 2
What can we discover that is true about a triangle? One well-known result concerns the sum of
the angles of a triangle. We will do this with a paper triangle in class … a sort of kinesthetic
proof. In a more formal way, we will work through a written proof.
Given: Δ ABC
Prove: m< 1 + m< 2 + m< 3 = 180


Draw ECD || AB
1.
2.
< 2  ______
2.
3.
< 3  ______
3.
4.
< ____ is supp to < ACD
4.
5.
m < ____ + m < ACD = 180
5.
Def supp
6.
m< 1 +m < ____ = m < ACD
6.
angle addition postulate
7.
m < 1 + m < ____ + m < ____= 180 7 .
8.
m< 1 + m < 2 + m < 3 = 180
Proof: 1.
8.
QED
Is the parallel postulate necessary in this proof? (We use this as the parallel postulate: If
corresponding angles are congruent, then lines are parallel.) As an implication of this fact, we
accept as true the fact that there is only one line through a point which is parallel to a given line.
We will discuss this assumption in more detail in a few weeks.
Next, we will look at other angles associated with triangles. An exterior angle is an angle which
is adjacent to one of the interior angles of a triangle. It is formed by extending one of the line
segment sides of a triangle.
Topic 11 (Triangles and Polygons)
Sketchpad Investigation:
1.
2.
Draw a triangle as shown.
Measure angles as needed to answer these questions,
using Sketchpad evidence.
3.

Questions:
How is < 1 related to the interior angles of the triangle?

What is the sum of the three exterior angles in a triangle?

Can a triangle have ________
a)
b)
c)
d)
e)
3 obtuse interior angles
2 obtuse interior angles
1 obtuse interior angle
3 congruent, obtuse interior angles
2 congruent, obtuse interior angles
f)
g)
h)
i)
j)
3 obtuse exterior angles
2 obtuse exterior angles
1 obtuse exterior angle
3 congruent, obtuse exterior angles
2 congruent, obtuse exterior angles
page 3
Topic 11 (Triangles and Polygons)
page 4

4.
Draw a right triangle, ΔABC. Keep BC unchanged, but move A along AC . Describe
what happens to m< 1 , m< 2, m < 3, and m< 4.
5.
In the right triangles that you constructed above, can m< 1 in one of the right triangles be

twice as large as m< 1 in another one, as you move A along AC ? If so, compare the
lengths of AC in the two triangles?
One Proof
Look at the conclusion above about how one of the exterior angles of a triangle is related to the
two remote interior angles. Prove your conclusion.
Topic 11 (Triangles and Polygons)
page 5
Polygons …what are they, and how do we use what we
already know to learn about their properties?
To produce polygons, we need to have some ground rules:
 The figure has to be bounded by line segments (but the segments do not all have to be
congruent), and the line segments can only intersect at their endpoints.
 The figure must be closed. (There has to be an interior and an exterior? How do we
know which is the interior?)
 Only two segments can intersect at each endpoint. (If more than two segments are drawn
at a vertex, then the excess over two may be “diagonals”.)

There are convex polygons and concave polygons.
Do some research to find the names of polygons. They are labelled according to the number of
sides that they have.
N
Name
Second name
(maybe)
1
2
3
4
5
6
7
8
9
10
11
12
13
20
40
52
73
n
Here are some more names …
3 - triangle
4 - quadrilateral
Topic 11 (Triangles and Polygons)
page 6
5 - pentagon
6 - hexagon
7 - heptagon
8 - octagon
9 - nonagon or enneagon
10 - decagon
11 - hendecagon or undecagon
12 - dodecagon
13 - triskaidecagon
14 - tetrakaidecagon
15 - pentakaidecagon
16 - hexakaidecagon
17 - heptakaidecagon
18 - octakaidecagon
19 - enneakaidecagon
20 - icosagon
21 - icosikaihenagon or henicosagon
Notice that “21” is ICOSA” KAI “HENAGON which means 20 (“icosa”) and (“kai”) 1
(“hen”) GON
22 - icosikaidigon or docosagon
23 - icosikaitrigon or tricosagon
24 - icosikaitetragon or tetracosagon
25 - icosikaipentagon or pentacosagon
Notice that “25” is ICOSA” KAI “PENTAGON which means 20 (“icosa”) and (“kai”)
5 (“penta”) GON
26 - icosikaihexagon or hexacosagon
27 - icosikaiheptagon or heptacosagon
28 - icosikaioctagon or octacosagon
29 - icosikaienneagon, enneacosagon or nonacosagon
30 - triacontagon
40 - tetracontagon
50 - pentacontagon
60 - hexacontagon
70 - heptacontagon
80 - octacontagon
90 - nonacontagon or enneacontagon
100 - hectagon
1000 - chiliagon
10000 - myriagon
100000 - decemyriagon
1000000 - hecatommyriagon
Source(s):http://www.2dshapes.com/polygons.html
Topic 11 (Triangles and Polygons)
page 7
Look up the definitions of equilateral and equiangular. Look up the definition of a regular
polygon.
Draw an equilateral but non- equiangular quadrilateral.
Draw an equiangular but non- equilateral quadrilateral.
Draw an equilateral and equiangular quadrilateral.
Draw an equilateral but non- equiangular hexagon.
Topic 11 (Triangles and Polygons)
page 8
Draw an equiangular but non- equilateral hexagon.
Draw an equilateral and equiangular hexagon.
Draw a regular triangle.
Draw a regular quadrilateral.
Draw a regular pentagon
Draw a regular hexagon.
Topic 11 (Triangles and Polygons)
page 9
Looking at the angles in polygons:
Draw a quadrilateral. Make it scalene and not regular for now. Pick one vertex, and draw all of
the diagonals from that vertex. (A diagonal is a line segment which connects non-consecutive
vertices.)
How many triangles are formed?
How many degrees are in the sum of the angles of each triangle?
Therefore, what is the sum of the measures of the interior angles of the quadrilateral?
Do you think that result would be different had you drawn a parallelogram? A square? A
rhombus?
Draw a pentagon. Make it scalene and not regular for now. Pick one vertex, and draw all of the
diagonals from that vertex.
Topic 11 (Triangles and Polygons)
page 10
How many triangles are formed?
How many degrees are in the sum of the angles of each triangle?
Therefore, what is the sum of the measures of the interior angles of the quadrilateral?
Do you think that result would be different had you drawn a regular pentagon?
Continue the process for other polygons. You are thinking inductively, looking for a pattern.
N
3
4
5
6
7
8
9
10
n
Name
Sum of interior angles
Topic 11 (Triangles and Polygons)
page 11
Open a Sketchpad document. Draw a convex quadrilateral. Try not to make it too special.
Extend each side to form one exterior angle, as shown. Measure each of the four exterior angles,
and calculate the sum of these exterior angles. Contort your quadrilateral and observe what
happens to the sum of the exterior angles.
Open a new sketch. Do the same investigation with a convex pentagon.
In a regular polygon, since the angles all have equal measure, then the measure of each angle can
be calculated by dividing the sum of the interior angles by the number of sides.
1.
Calculate the size of each interior angle in a regular pentagon. Show your work.
2.
Calculate the size of each interior angle in a regular hexagon. Show your work.
Topic 11 (Triangles and Polygons)
page 12
3.
Calculate the size of each interior angle in a regular nonagon. Show your work.
4.
Calculate the size of each interior angle in a regular 42-gon. Show your work.
The same idea is used to find the size of each exterior angle in a regular polygon.
5.
Calculate the size of each exterior angle in a regular pentagon. Show your work.
6.
Calculate the size of each exterior angle in a regular hexagon. Show your work.
7.
Calculate the size of each exterior angle in a regular nonagon. Show your work.
Summarize the calculation of these angles for an n-gon:
Sum of the measures of the interior angles =
Measure of each interior angle (regular polygons) =
Topic 11 (Triangles and Polygons)
page 13
Sum of the measures of the exterior angles =
Measure of each exterior angle (regular polygons) =
Design an Excel spreadsheet to see the patterns in the angles in polygons. (You will be using
your laptop to this.) The first rows should look like this, and you will be shown how to enter
formulas in a spreadsheet.
gles in Polygons of es of m of m of angles <'s t <'s 3 1
180
4 2
360
5 3
540
6 4
720
7 5
900
8 6
1080
9 7
1260
10 8
1440
11 9
1620
12 10
1800
360
360
360
360
360
360
360
360
360
360
GULAR ch int gle 60
90
108
120
128.5714
135
140
144
147.2727
150
ch ext gle 120 90 72 60 51.42857 45 40 36 32.72727 30 Observations:
 What happens in the sequence of the sum of the interior angles?

What happens in the sequence of the sum of the exterior angles?
Topic 11 (Triangles and Polygons)
page 14

What does the size of each interior angle (regular polygons) sem to approach as n gets
really large?

What does the size of each exterior angle (regular polygons) sem to approach as n gets
really large?

How many different kinds of regular polygons have exterior angles which are integers?

How many different kinds of regular polygons have interior angles which are all less than
1510?
How Big are the ANGLES in TRIANGLES &
POLYGONS
Here are some triangles and polygons whose angles need to be calculated. Use the new formulas
that you have learned and verified.
1.
Verify that the sum of the exterior angles, taken one at each vertex, of any n-gon is 3600.
(This can be done in an algebraic flavor.)
Topic 11 (Triangles and Polygons)
page 15
2.
The measures of the three angles of a triangle are in the ratio 2:4:9. Find the measure of
each angle. (Hint: Call the measures of the angles 2x, 4x, and 9x).
3.
The measures of the three angles of a triangle are in the ratio 2:3:5. Find the measure of
the smallest exterior angle.
4.
Four of the angles in a pentagon measure 45, 65, 128, and 147. What is the measure of
the fifth angle?
5.
Two of the interior angles of a quadrilateral have measure 360 and 840. The other two
angles are in the ratio 3:5. Find the measure of the other two interior angles as well as the
second to largest exterior angle.
6.
The measure of one of the acute angles of a right triangle is 150 less than twice the
measure of the other acute angle. Find the measure of each of the acute angles.
Topic 11 (Triangles and Polygons)
7.
Find the measures of the numbered angles in the drawing. It is given that AB  BC .
m<1=
m<2=
m<3=
m<4=
8.
page 16
Okay, not all numerical problems …..
Given: The plane figure
Prove: r + s = t + q
Topic 11 (Triangles and Polygons)
page 17
9.
Each of the five angles of a convex hexagon measures 1300. What is the measure of the
sixth interior angle?
10.
How many sides does a regular convex polygon have if one exterior angle has measure
240 ?
11.
What is the measure of an interior angle of a regular polygon with 12 sides?
12.
If each interior angle of a regular polygon has a measure of 1700, how many sides does it
have?
13.
Is it possible for a regular polygon to have interior angles whose measure is _____?
(Describe why or why not.)
a) 100
d)
500
b)
450
c)
0.50
Topic 11 (Triangles and Polygons)
page 18
14.
If each exterior angle of a regular polygon has measure 300, what is the sum of the
measure of the interior angels of that polygon?
15.
Discover a formula for the total number of diagonals that can be drawn from all vertices
in a convex n-gon.
16.
One angle of a triangle has measure 450, and the other two angles are in the ratio 1:2.
Find the other two angles.
17.
Find the angle between the two bisectors of the acute angles of a right triangle.
18.
An exterior angle of a triangle has measure 1300. One remote interior angle is 360 more
than the other remote inter angle. Find the measures of those remote interior angles.
Topic 11 (Triangles and Polygons)
page 19
19.
The sum of an angle and twice its complement is 1460. Find the measure of the angle.
20.
Find the first instance where an n-gon and an n+1-gon contribute one interior angle each
whose sum exceeds 2700. (Assume that the polygons are sitting adjacent to each other,
and they have congruent sides.)
21.
Write three examples of combinations of polygons whose interior angles share a vertex
and yet do not intersect so that the sum of one each of their interior angles does not
exceed 3600.
ISOSCELES TRIANGLES…..
Two angles of an isosceles triangle are congruent. (Accept this right now without proof. The
verification will come in a few weeks.).
The congruent angles are called base angles,
and the third angle is called the vertex angle.
The congruent sides are called legs, and the
third side is called the base.
Vertex angle
leg
leg
base angle
base angle
base
Topic 11 (Triangles and Polygons)
22.
m < C = 1000
m< A = _____
m< B = _____
23.
m < C = 460
m< A = _____
m< B = _____
24.
m < 1 = 1100
m< 2 = _____
m< 3 = ______
m < 4 = ______
25.
m < 4 = 820
m< 1 = _____
m< 2 = ______
m < 3 = ______
26.
m < 2 = 3x + 10
m < 4 = 2x + 40.
Solve for x.
page 20
Topic 11 (Triangles and Polygons)
27.
m < 1 = ______
28.
x = ________
y = ________
(regular octagon)
29.
Find the measures of the numbered angles.
(regular pentagon)
page 21
Topic 11 (Triangles and Polygons)
page 22
30.
The supplement of an angle is 10 0 more than three times its complement. Find the angle.
31.
Two angles in a triangle sum to 2
3 of the measure of the third angle. One of those two
angles is 20 more than the other. Find the measure of the largest angle in the triangle.
Problems are from Mathematics Review Exercises (Smith and Fagan, Ginn and Company, 1961), Modern School
Mathematics: Geometry (Jurgenson, Donnelly, Dolciani,Hougling Mifflin, 1972), and your teacher on a sunny, cold
Friday afternoon.
Definition: A tessellation is a plane filling combination of shapes. (There are no gaps nor any
overlaps of the shapes).
This means that the interior angles that surround a point have to sum to 360 exactly. For
example, four squares can surround a point. Since each square has an interior angle with
measure 900, then the sum of four of these with be exactly 3600. It is also true that two regular
octagons and one square will surround a point. This is because the interior angles are 135, 135,
and 90, and 135 + 135 + 90 = 360.
A tessellation is
called a regular
Topic 11 (Triangles and Polygons)
page 23
tessellation if only one kind of regular polygon is used (such as four squares), and the symbol is
4-4-4-4 (for square-square-square-square). A tessellation is called a semi-regular tessellation if
different kinds of regular polygons are used, but the configuration at every vertex is identical
throughout the tiling. The tessellation with two octagons and one square above is an example,
and its symbol is 8-8-4 (or 8-4-8 or 4-8-8: all are the same). A demi-regular tessellation is
one made of different kinds of regular polygons are used, but the configuration at various
vertices is not identical throughout the tiling.
Make a list of all of the regular tessellations, using symbols like 4-4-4-4. Use the cut out tiles to
experiment.
Make a list of all of the semi-regular tessellations. Use the cut out tiles to experiment.
Make a list of all of the demi-regular tessellations. Use the cut out tiles to experiment.
Topic 11 (Triangles and Polygons)
Here are some more complex tessellations. Describe what happens at various vertices.
page 24
Topic 11 (Triangles and Polygons)
page 25
Tessellations are from http://library.thinkquest.org/16661/templates/index.html .
Remember that a tessellation is a tiling of the plane. The shapes that make up the tessellation
have to be able to cover the plane with go gaps and no overlaps. (The tiling has to be able to
continue indefinitely.) Our previous work used only regular polygons to do the tilings. Now, we
will examine whether there are other shapes that can tile.
QUESTION 1:
Will all triangles tile? If not, which ones will tile?
QUESTION 2:
Will all quadrilaterals tile? If not, which ones will tile?
QUESTION 3:
Will all pentagons tile? If not, which ones will tile?
QUESTION 4:
What kinds of irregular shapes will tile?
Topic 11 (Triangles and Polygons)
QUESTION 1:
page 26
Will all triangles tile? If not, which ones will tile?
Use the example of this one triangle, from which we will generalize to other triangles. Use a
template of the triangle, and figure out how to position it, repeatedly, on the grid to tile.
USE PENCIL!! You might need to erase!
QUESTION 2:
Will all quadrilaterals tile? If not, which ones will tile?
Try one example of various kinds of quadrilaterals to see if they will tile. If they do indeed tile,
look for a strategy to determine how they do so. (Is it already clear that rectangles and squares
will tile???) USE PENCIL!! You might need to erase!
Parallelograms
Topic 11 (Triangles and Polygons)
Trapezoids
Non-Convex Quadrilaterals
page 27
Topic 11 (Triangles and Polygons)
QUESTION 3:
page 28
Will all pentagons tile? If not, which ones will tile?
Choose a pentagon … not a regular pentagon (because we already know what happens)
and try to tile with it.
Another pentagon example
Topic 11 (Triangles and Polygons)
QUESTION 4:
What kinds of irregular shapes will tile?
Escher style tessellations
M. C. Escher was a Dutch artist whose specialty was tessellations.
•
•
•
•
•
Born June 17, 1898 in the Dutch province of Friesland.
He began studying tilings around 1935.
He created his most famous tessellation, “Reptiles” in 1943.
Escher drew his last tessellation in July of 1969.
He died at the age of 73 in March of 1972.
You would enjoy seeing some of his work. Look it up online!
Here are some instructions to produce an Escher style tessellation
From: The Geometer's Sketchpad and Tessellations
Cathi Sanders
Punahou School, Honolulu, Hawaii
Construct a parallelogram, using Parallel Line in
the Construct menu.
Construct a random shape on the left side of the
parallelogram.
page 29
Topic 11 (Triangles and Polygons)
Select D and then C, and Mark Vector "D->C" in
the Transform menu. Then select all of the random
shape and Translate it By Marked Vector using the
Transform menu. (Vector is the name for a segment
with a given length and a given direction.)
Construct a 2nd random shape on the top of the
parallelogram.
Mark D to A as a vector, and translate the 2nd
random shape.
Select the vertices of your polygon in consecutive
order and Construct Polygon Interior
page 30
Topic 11 (Triangles and Polygons)
page 31
Translate the whole polygon (vertices, sides and interior) by a vector DC and then by a vector
DA to create a tessellation. Use different colors for adjacent polygons. Drag any point to change
your tessellation.
Finally …. Experiment with Sketchpad to produce a creative tiling. Make it colorful, perhaps
intricate, certainly interesting and visually appealing. Use the attached instructions from the
Exploring Geometry with The Geometer’s Sketchpad © 2002 Key Curriculum Press to help
you.