Polygons and their Angle Measures

Essential Questions

   How do I identify and classify polynomials?

How do I find the measures of interior and exterior angles of polygons?

How do I use measures of angles of polygons to solve problems?

 

Polygon- a plane figure that (1) is formed by 3 or more segments, such that no two sides with a common endpoint are collinear and (2) each side intersects exactly 2 other sides , 1 at each endpoint.

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Means the figure is closed and no sides cross over each other.

Vertex- the endpoints of each side

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Think of them as the corner points!

Example

vertex side

Example Is the figure a polygon?

yes no yes no

Number of sides   3 4     5 6 7 8   9 10   12 n Type of polygon  Triangle   Quadrilateral Pentagon  Hexagon   Heptagon Octagon  Nonagon   Decagon Dodecagon  N-gon

Special types of Polygons

     Convex- no line that contains a side of a polygon goes through its interior Concave- opposite of a convex Equilateral- all sides are  Equiangular- all angles are  Regular polygon- equilateral and equiangular

 Diagonal- a segment that joins 2 nonconsecutive vertices

A segments AE and AD C are diagonals B

j

D E

Interior Angles of a Quadrilateral  The sum of the measures of the interior  ‘s

A

o .

A 1 2 B m



1 + m



2 + m



3 + m



4 = 360 o D 4 3 C

Example Find x

X 55 o

 x+x+55+55=360  2x+110=360   2x=250 X=125

X

Polygon Interior Angles Theorem  The sum of the measures of the interior angles of a convex n-gon is (n-2) •180°.

Corollary to the Polygon Interior Angles Theorem  The measure of each interior angle of a regular n-gon is (n 2)  180 .

n

Example

 Find the value of x in the diagram shown. 108 °+146°+101°+113°+153°+x°=720° 621 + x = 720 108  x = 99

The measure of the 6 th interior angle is 99 °!

x  146  153  101  113 

Try This!

 Find the value of x.

93 º+123º+102º+100º+xº = 540º 418 + x = 540 93 

x = 122

x  123  100  102 



Example

The measure of each interior angle of a regular polygon is 140 °. How many sides does the polygon have?

(n  2)  180  140 o n 180(n  2)  140n 180n  360 40n n   9 360  140n

The polygon has 9 sides.

Try This!

 The measure of each interior angle of a regular polygon is 165 º. How many sides does the polygon have?

(n  2)  180  165 n 180(n  2)  165n 180n  360  165n 15n  360 n  24

The polygon has 24 sides.

Polygon Exterior Angles Theorem  The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360 º.

3 2 4 1 5

Corollary to the Polygon Exterior Angles Theorem  The measure of each exterior angle of a regular n gon is… 3 360 o n 2 4 1 5

Example

 Find the value of x.

2x  +2x  +4x  +3x  +x  = 360  12x = 360 x = 30 4x  2x  2x  x  3x 

Example

 Find the value of x.

x = 360/7 x  51.43

º x 

Try This!

 Find the value of y.

2y  y  + y  + 2y  + 2y  = 360  6y = 360 y = 60 y  y  2y 

Summarizer

 Explain in words how to find the measure of each interior angle and each exterior angle in a regular polygon.