Download Section 8.4

Chapter 8
Introductory Geometry
Section 8.4
Angle Measures of Polygons
Angle Measures of Polygons
When the angle measures of polygons are discussed what is being referred to are
the measures of the interior angles of the polygons. Individually these angles can
have any measures, but when you add the measures of all the angles they can
only be a certain number that depends on the number of sides. For example, in a
triangle the measure of any angle can be any number between 0 and 180, but if
you add the interior angles together they add up to 180.
2
1
2
m1 + m2 + m3 = 180
3
1
4
3
m1 + m2 + m3 + m4 = 360
Angle Measures of Triangles
One way to see that the angles of a triangle
combine to give a straight angle (i.e.
measure 180) is to make three congruent
copies of the triangle and put them together
as pictured to the right. Notice the sides form
a straight line.
1
2
3
2 1
3
1
2
3
m1 + m2 + m3 = 180
The way this is established formally with
deductive reasoning (formal deduction van Hiele
level 4) is by using the principle of alternate
interior angles. Given a triangle construct a line
parallel to one side going through the vertex on
the opposite side.
2  4 and 3  5 (Alternate Interior Angles)
4 1
2
5
3
m1+m2+m3 = m1+m4+m5 = 180
The result that the measures of interior angles of triangles is 180 form the basis
for finding the interior measure of the angles of all the other polygons. This is
done by breaking up the other polygons into triangles and looking at the angles
of the polygons as the angles of triangles.
Quadrilaterals
Each of the quadrilaterals below is broken into two triangles by inserting a purple
line segment in each one of them.
In the quadrilateral to the right that has been
broken into two triangles we add up all the
interior angles and rearrange them into two
triangles
2
3
m1+ m2+ m3+ m4+ m5+ m6
4
= (m1+ m2+ m3)+ (m4+ m5+ m6)
1
= 180 + 180
6
5
= 360
This is another one of the patterns that exist within quadrilateral shapes is that the
sum of the interior angles is always the same (like the number of diagonals). In
fact, the interior angle sum of quadrilaterals is always 360.
What about other shapes? Polygons that have more sides than 4. If the polygon
can always be broken apart into the same number of triangles the sum of the
interior angles is always the same. Below are some examples of pentagons.
3 triangles
3 triangles
3 triangles
3 triangles
The interior angle sum for a pentagon can be broken apart in
a similar way as a quadrilateral except you have 9 angles
instead of 6.
5
6
7
8
4
m1+ m2+ m3+ m4+ m5 + m6 + m7 + m8+ m9
=(m1+ m8+ m9)+ (m3+ m4 + m5) + (m2 + m6+ m7)
3
2
1
9
= 180 + 180 + 180
= 540
The interior of a pentagon can always be broken into 3 triangles. A pentagon’s
interior angle sum is the interior angles sum of 3 triangles which is 540.
Hexagons can always be broken into 4 triangles. The interior angle sum will be the
interior angle sum of 4 triangles.
Interior angle sum of a hexagon
= Interior angle sum of 4 triangles
= 180 + 180 + 180 + 180
= 4 · 180
= 720
4 triangles
4 triangles
We will use inductive reasoning to see if we can find a pattern using the entries in
the table below.
Name of
Shape
Number of
Sides
Number of
Triangles
Sum of
Angles of
Triangles
Sum of
Angles of
Shape
Triangle
3
1
1·180
180
Quadrilateral
4
2
2·180
360
Pentagon
5
3
3·180
540
Hexagon
6
4
4·180
720
Heptagon
7
5
Octogon
8
6
“n-gon”
n
n-2
5·180

6·180

(n2)·180
900
1080
(n2)·180
The sum of the interior angles of a polygon with n sides is: (n-2)·180
Regular Shaped Polygons
A polygon is called regular if all of its sides are congruent to each other and all of
its interior angles are congruent to each other. A few regular shapes you know
already.
A regular triangle is called an equilateral triangle.
A regular quadrilateral is called a square.
regular pentagon
regular hexagon
regular octagon
Interior Angles of Regular Shaped Polygons
Since each angle of a regular shaped polygon has the exact same measure we
can find the measure of an angle by dividing the total angle sum by the number
of angles which is also the number of sides. A regular polygon with n sides will
have each of its angles measuring the following:
sum of all angles
(n  2) 180

number of angles
n
The formula on the previous slide cab be applied to equilateral
triangles. The value of n=3 and we get the following:
(3  2) 180 1180 180


 60
3
3
3
The formula on the previous slide cab be applied to squares.
The value of n=3 and we get the following:
(4  2) 180 2 180 360


 90
4
4
4
How can the formula be applied to find the interior angles of
the regular hexagon picture to the right?
(6  2) 180 4 180 720


 120
6
6
6
60
60
60
90
90
90
90
120 120
120
120
120 120