Download 5A Interior Angles in Polygons

5A Interior Angles in Polygons
Name______________________________________
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Polygon- _________________________________________________________________________________________________________________________________
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Convex- __________________________________________________________________________________________________________________________________
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Concave- _________________________________________________________________________________________________________________________________
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Regular- __________________________________________________________________________________________________________________________________
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1.
CENTRAL ANGLE: The regular polygon is inscribed in a circle whose central angle measure is 360o.
Use the polygons and central angle measurements to complete the following table.
Regular Polygon
# of Sides
Central Angle
Measure
Triangle
Quadrilateral
Pentagon
Hexagon
n-gon
a.
Each regular polygon is divided into triangles. What type of triangles are they? Why?
b.
For each regular polygon, what is the relationship between these triangles? Why?
c.
Use the pattern in the table to find the central angle measure of a regular octagon (8 sides).
2.
Joshua created a table to explore the relationship between the number of triangles and the interior
angle measurements in regular polygons. Complete Joshua’s table. *FYI when breaking a polygon
into triangles the diagonal segments are drawn from a single vertex to form each triangle.*
Sum of
Interior
Regular
# of
# of
# of Triangles
Interior
Angle
Polygon
Sides
Triangles (180°) =
Angles
Measure
Triangle
Quadrilateral
Pentagon
Hexagon
n-gon
3.
Use the pattern in the table to find the interior angle measure of a regular decagon (10 sides).
4.
Madeline created a different table to explore the relationship between the number of triangles and the
angle measurements in regular polygons. Complete Madeline’s table.
Regular
# of Sides # of Triangles
# of Sides (180°) – 360° =
Polygon
Sum of Interior
Angles
Triangle
Quadrilateral
Pentagon
Hexagon
n-gon
5.
When the number of sides in a regular polygon increases by 1, why does the interior angle sum increase
by 180°?
6.
Use the pattern in the table to find the sum of the base angles of an isosceles triangle drawn from the
center of a regular nonagon (9 sides).
7.
When congruent isosceles triangles are drawn from the center of a regular polygon, why is the base
angle sum of any one of the isosceles triangles equivalent to the measure of an interior angle of the
polygon?
8.
The interior angle sum can be calculated using Joshua’s expression (n – 2)180, or 180n – 360 (as done
by Madeline). What is the relationship between these two expressions?
Unless otherwise noted, determine the measure of one interior angle in each polygon. Round your answer to the nearest
tenth if necessary.
9. What is the value of x?
10.
11.
12. What is the value of x?
13. regular 24-gon
14. regular quadrilateral
15. regular 23-gon
16. regular 16-gon
17.
18.
19. What is the value of
20.
x?
21. regular quadrilateral
22. regular 18-gon
25. Is there a regular polygon with an interior angle sum
of 9000°? If so, what is it?
26. How many sides does a polygon have if the sum of its
interior angles is 2340°?
23. regular dodecagon
24. regular 15-gon
27. What are the values of w, x, and y?