Chapter 8 Introductory Geometry Section 8.4 Angle Measures of Polygons.
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Transcript of Chapter 8 Introductory Geometry Section 8.4 Angle Measures of Polygons.
![Page 1: Chapter 8 Introductory Geometry Section 8.4 Angle Measures of Polygons.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649dd95503460f94ace2de/html5/thumbnails/1.jpg)
Chapter 8
Introductory Geometry
Section 8.4
Angle Measures of Polygons
![Page 2: Chapter 8 Introductory Geometry Section 8.4 Angle Measures of Polygons.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649dd95503460f94ace2de/html5/thumbnails/2.jpg)
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.
1
2 3
m1 + m2 + m3 = 180
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.
m1 + m2 + m3 + m4 = 360
1
23
4
1
2 1 2
2 1
3
3
3
m1 + m2 + m3 = 180
![Page 3: Chapter 8 Introductory Geometry Section 8.4 Angle Measures of Polygons.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649dd95503460f94ace2de/html5/thumbnails/3.jpg)
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)
m1+m2+m3 = m1+m4+m5 = 180
1
2 3
4 5
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.
![Page 4: Chapter 8 Introductory Geometry Section 8.4 Angle Measures of Polygons.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649dd95503460f94ace2de/html5/thumbnails/4.jpg)
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
m1+ m2+ m3+ m4+ m5+ m6
= (m1+ m2+ m3)+ (m4+ m5+ m6)
= 180 + 180
= 360
1
2
3
4
56
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
![Page 5: Chapter 8 Introductory Geometry Section 8.4 Angle Measures of Polygons.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649dd95503460f94ace2de/html5/thumbnails/5.jpg)
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.
m1+ m2+ m3+ m4+ m5 + m6 + m7 + m8+ m9
=(m1+ m8+ m9)+ (m3+ m4 + m5) + (m2 + m6+ m7)
= 180 + 180 + 180
= 540
1
23
4
56
7
8
9
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
![Page 6: Chapter 8 Introductory Geometry Section 8.4 Angle Measures of Polygons.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649dd95503460f94ace2de/html5/thumbnails/6.jpg)
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
Octogon
“n-gon”
7 5 5·180 900
n n-2 (n-2)·180 (n-2)·180
8 6 6·180 1080
The sum of the interior angles of a polygon with n sides is: (n-2)·180
![Page 7: Chapter 8 Introductory Geometry Section 8.4 Angle Measures of Polygons.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649dd95503460f94ace2de/html5/thumbnails/7.jpg)
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:
n
n
anglesofnumber
anglesallofsum
180)2(
![Page 8: Chapter 8 Introductory Geometry Section 8.4 Angle Measures of Polygons.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649dd95503460f94ace2de/html5/thumbnails/8.jpg)
The formula on the previous slide cab be applied to equilateral triangles. The value of n=3 and we get the following:
603
180
3
1801
3
180)23(60
60
60
The formula on the previous slide cab be applied to squares. The value of n=3 and we get the following:
904
360
4
1802
4
180)24(
90 90
90 90
How can the formula be applied to find the interior angles of the regular hexagon picture to the right?
1206
720
6
1804
6
180)26( 120
120 120
120
120 120