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Topic 3Angle Properties in Polygons
Unit 2 Topic 3
Explore
Polygon Number of Sides
Diagram
Number of Triangles
Sum of Angle Measures
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Complete the following chart. What do you notice?
ExplorePolygon Number
of SidesDiagram
Number of Triangles
Sum of Angle Measures
Triangle 3 1 180⁰
Quadrilateral 4 2 360⁰
Pentagon 5 3 540⁰
Hexagon 6 4 720⁰
Heptagon 7 5 900⁰
Octagon 8 6 1080⁰
Can you come up with a way to calculate the sum of the angle measures based on the number of sides in the polygon?
Information
•You should notice that
▫The sum of the angle measures in the polygon is equal to 180⁰ X (number of sides – 2)
Example 1Determining the sum of the measure of the interior angles.
Determine the sum of the measures of the interior angles of a
a) hexagon b) 15-sided polygon (pentadecagon)
Try this on your own first!!!!
Example 1: Solution
a) hexagon b) 15-sided polygon (pentadecagon)
180 ( 2)
180 (6 2) 720
n 180 ( 2)
180 (15 2) 2340
n
A hexagon has 6 sides.
Example 2Determining the measure of each interior angle
Determine the measure of each interior angle of a regular:
a) pentagon b) octagon
Try this on your own first!!!!
Example 2: Solution
a) pentagon b) octagon
8 sides5 sides
180 ( 2)
180 (5 2) 540
540 5 108
n
180 ( 2)
180 (8 2) 1080
1080 8 135
n
5 equal angles 8 equal angles
Example 3Determining how many sides
Determine how many sides a polygon must have if its interior angles add up to:
a) 3060⁰ b) 1800 ⁰
Try this on your own first!!!!
a) 3060⁰ b) 1800 ⁰
180 ( 2) 3060
2 17
19
n
n
n
180 ( 2) 1800
2 10
12
n
n
n
Example 3: Solution
Example 4Determining the missing angles
a) Determine the value of x.
b) Determine the value of each internal angle.
Try this on your own first!!!!
A
B
CD
4x
3x + 20
2x + 40
x + 20
Example 4: Solution
a) Determine the value of x.
b) Determine the value of each internal angle.
A
B
CD
4x
3x + 20
2x + 40
x + 20
(3 20) (2 40) (4 ) ( 20) 360
3 20 2 40 4 20 360
10 80 360
10 280
28
x x x x
x x x x
x
x
x
3 20
3(28) 20 104
x
2 40
2(28) 40 96
x
4
4(28) 112
x
20
28 20 48
x
Example 5Floor tile designs
Try this on your own first!!!!
A floor tiler designs custom floors using tiles in the shape of regular polygons. Can the tiler use regular octagons and squares to tile a floor, if they have the same side length?
Hint: Try drawing a picture.
Example 5: Solution
Etc, etc…
Yes he can.
Example 6Determining missing angles
Try this on your own first!!!!
Determine the values of a, b, c, and d.
a
b
c
d
Example 6: Solution
a
b
c
d
1. We have a regular hexagon. The sum of all the interior angles is:
The value of each interior angle is:
180 ( 2)
180 (6 2) 720
n
720 6 120 120⁰
120⁰
2. We can find the blue angles by noting that they lie on a straight line with 120⁰, making them supplementary.
60⁰
60⁰
180 120 60
3. All three angles in a triangle must add to 180⁰, so the third angle must be 60⁰.
180 60 60 60
60⁰
3. Using corresponding angles, angle d is 60⁰.
180 120 60
60⁰
Need to Know:• The sum of the measures of the interiors angles of a
polygon with n sides is determined using .
• To determine the measure of each interior angle in a regular polygon, divide the sum of the interior angles by the number of angles (same as the number of sides).
• The sum of the measures of the interior angles of a polygon is 360. You’re ready! Try the
homework from this section.
180 ( 2)n
