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Properties and Attributes of Polygons

6.1

 

Vocabulary:

 

Side of a Polygon- A segment that forms part of a polygon

 

Vertex of a polygon- A common endpoint of two sides

 

Diagonal- A segment that connects two nonconsecutive vertices

 

Regular polygon- A polygonwith congruent sides and angles

 

Concave- Any part of a diagonal that has points outside the polygon

 

Convex- A diagonal with no points outside the polygon

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--Example 1: Identifying polygons

 

Tell if the figure is a polygon. If it is, name it by the number of sides it has.

Yes. Hexagon

 

Not a polygon

 

--Example 2: Classifying polygons

Tell if polygons are regular or irregular. Tell if it is concave or convex.

 

 Regular. Convex

 

Irregular. Concave

 

Theorem 6-1-1 Polygon Angle Sum Theorem The sum of the interior angles on a convex polygon with n sides is (n-2)180 degrees.

 

 

--Example 3: Finding Interior Angle Measures And Sums In A Polygon

 

 

 

-A: Sum of Interior Measures of a Convex Hexagon

 

(n-2)180  --  Polygon Angle Sum Theorem

 

(6-2)180  --Hexagons have six sides. Substitute 6 for n.

 

720 --  simplify

 

 

 

-B: Find measure of each interior angle of a regular octagon.

 

(n-2)180  -- Polygon Angle Sum Theorem

 

(8-2)180=1080  --  Substitute 8 for n and simplify

 

1080/8=135  --  divide by eight to find the measure of one interior angle

 

 

 

-C: Find measure of interior angle of Pentagon ABCDE.

(not drawn to scale)

 

(5-2)180=540  --  Polygon Angle Sum Theorem

 

A+B+C+D+E=540  --  Polygon Sum Theorem

 

3x+x+x+3x+x=540  --  Substitute

 

9x+540  --  Combine Like Terms

 

x=60  --  divide both sides by nine

 

B=C=E=60

 

A=D=3(60)=180

 

 

 

Theorem 6-1-2 Polygon Exterior Sum Theorem   The sum of exterior angle measures,one angle at each vertex, of a convex polygon=360 degrees.

 

 

--Example 4: Find Exterior Angle Measures In Polygon

 

-A: Find measures of each exterior angle of a regular quadrilateral

 

 

The quadrilateral has four sides and angles

 

The sum of the exterior angles is 360

 

The measure of one exterior angle is 360/4=90

 

The measure of each exterior angle of a regular quadrilateral is 90.

 

 

 

-B: Find Value of g in polygon WXYZ

 

 

 

g+g+5g+3g=360  --  Polygon Exterior Sum Theorem

 

10g=360  --  Combine Like Terms

 

g=36  --  Divide by 10

 

 

 

--Example 5: Real-World Application

 

The folds on the lid of a gift box form a regular hexagon. Find each exterior measure.

 

 

 

 

 

A regular hexagon has 6 congruent angles, so divide sum by 6.

 

 

 

Angle LKM=360/6=60

 

Angle LKM is an exterior angle of a regular hexagon. By the Polygon Exterior Angle Sum Theorem, the sum of an exterior angle measures is 360 degrees.

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Midterm question: 

 

 

What is the sum of the interior measures of a ragular conves nonagon?

(A nonagon has nine sides)

 

 

(n-2)180

(9-2)180

(7)180

1260 degrees