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An exterior angle is formed by a side of the triangle and an extension of another side.

 

Remote Interior angles are the 2 angles inside the triangle furtherest  from the indicated exterior angle. (NON-ADJACENT TO THE EXTERIOR ANGLE)  Place your fingertip on the exterior angle at its vertex. You are touching the exterior angle and its adjacent angle.  You are NOT touching the remote interior angles.
remoteinterior.jpg

Angle 5  and  Angle 6 are remote interior to the exterior  Angle 1 .

 Angle 4  and  Angle 6 are remote interior to the exterior Angle 2.

Angle 4  and  Angle 5 are remote interior to the exterior Angle3.

Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the measures of the 2 remote interior angles.

Angle 1 is an exterior angle, so, its measure is equal to the sum of the measures of the 2 interior angles to which it is not adjacent, namely, Angle 5 and Angle 6.  (Still referring to the drawing above.)

 m Angle 1  =   m Angle 5 +m Angle 6

m  Angle 2  =   m Angle 4 +m Angle 6

m Angle 3  =   m Angle 4 +m Angle 5

 

polygons.jpg
A polygon is a closed plane figure formed by at least 3 line segments.
 
 
 

convexpolygon.jpg

A convex polygon's diagonals are completely within the polygon.

 
 
 

 

concave.jpg

Some portion of at least one of the  concave polygon's diagonals is outside the polygon. (Has a cave or indentation.)

 
 
 

regularpolygon.jpg

A regular polygon has

all congruent sides (equilateral) AND all congruent angles (equiangular).  NOTE: Polygons can be equiangular and NOT equilateral or equilateral and not equiangular.

 
 
 

Polygon Interior Angle-Sum Theorem:  The sum of the measures of the interior angles of a polygon is 180(n - 2), where n is the number of sides

 
 

Find x, y, and r.

polygonex1.jpg

Look at the triangle.

54 +48 + x = 180    solve for x

   102 +   x  = 180

  - 102           - 102

              x =  78

What does x and y form?      A linear pair. So:

x + y = 180  

          but you know x = 78 substitute 78 in for x

78+ y = 180

-78         -78

       y  =  102   

Now we need to find r, but first we need to find the unnamed angle in the polygon.

 

 

What does x and the unnamed angle(we will call it z now) in the polygon form??

Vertical Angles,  SO

x = z   but x is still 78 so

78 = z

Now you will set the actual sum of the angles in the polygon equal to the formula sum.

Count the number of sides of the polygon--5.   Use the sum of interior angles formula

 

sum of interior angles  of convex polygon=

180 (n - 2 ) 

Sub 5 in for n

 

 

sum of interior angles =180 (5 -  2) parenthesis 1st

sum of interior angles = 180 (3) 

sum of interior angles = 540

Now

actual sum of interior angles     = formula sum of interior angles

  115+ 120 + r + 100 + 78 we found = 540

           413       + r                            = 540

                                     r                 = 127

              x =  78,  y  =  102  , r = 127  FINALLY

One interior angle of a regular polygon shares the sum of the interior angles equally with the other angles.  So

1 interior angle of a regular polygon = sum divided by the number of angles (same as number of sides)   n

 

1 interior angle of a regular polygon =

180(n - 2)

    n

 

EX.  Find the measure of an interior angle of a dodecagon.

A dodecagon has 12 sides.

1 interior angle of a regular polygon =

180(n - 2)

    n

1 interior < of a dodecagon  =     

180(12 - 2)

     12

1 interior < of a dodecagon       =

180(10) ÷ 12

1 interior < of a dodecagon       = 

150.

 

 

EX.  A regular polygon has an interior angle of 135 degrees.

Name the polygon.

Use                             

      1 interior <  

        of a reg  =                 polygon

180(n - 2)

    n

but sub in 135 for left side.

                          135 =

180(n-2)      

      n

 

Multiply both sides by n OR place 1 under 135 and cross multiply  

                          135n =

180(n – 2)             

Distribute  180 to n and – 2                          

                          135n =

                       - 180n 

    –  45n   =         

  180n –  360        

 - 180n 

      – 360

Divide both sides by – 45        

                         n =

8

The polygon has 8 sides so it is an octagon.

 

 

Polygon Exterior Angle-Sum Theorem:  The sum of the measures of the exterior angles of a polygon (one at each vertex) is 360.

 

1 interior < of a regular polygon =

180(n - 2)

    n

 

EX.  Find the measure of an interior angle of a dodecagon.

A dodecagon has 12 sides.

1 interior < of a regular polygon =

180(n - 2)

    n

1 interior < of a dodecagon  =      

180(12 - 2)

     12

1 interior < of a dodecagon       =

180(10) ÷ 12

1 interior < of a dodecagon       =

150.

 

 

EX.  A regular polygon has an interior angle of 135 degrees.

Name the polygon.

Use                             

1 interior < of a regular polygon =

180(n - 2)

    n

but sub in 135 for left side.

                          135 =

180(n-2)       

      n

 

Multiply both sides by n OR place 1 under 135 and cross multiply  

                          135n =

180(n-2)              

Distribute                           

                          135n =

                       - 180n  

                       -   45n   =         

180n-  360         

 - 180n 

        - 360

Divide both sides by 45        

                               n =

8

The polygon has 8 sides so it is an octagon.

 

 

Polygon Exterior Angle-Sum Theorem:  The sum of the measures of the exterior angles of a polygon (one at each vertex) is 360.

If it is a regular polygon, each of its exterior angles will also be congruent, so they will share that 360 equally. The number of exterior angles (1 at each vertex) will be the same as the number of sides.

1 exterior < of a REGULAR polygon = 360 divided by the number of sides

 1 exterior < of a REGULAR polygon = 360/n

 

EX.  Find the measure of an exterior angle of a regular 20-gon.

1 exterior < of a REGULAR polygon = 360/n

1 exterior < of a REGULAR 20-gon = 360 ¸ 20

1 exterior < of a REGULAR 20-gon = 18 degrees

 

EX.  The measure of an exterior angle of a regular polygon is 20 degrees.  How many sides does it have?

Use

1 exterior < of a REGULAR polygon =           

360

   n              

but sub in 20 for the left side.

                                  20 =           

360

   n              

multiply both sides by n or cross-multiply after putting 1 under the 20.

                  20n             =

360

 

 

divide both sides by 20

                   n             =

18

 

The polygon has 18 sides.

 

REGULAR formulas ONLY work on REGULAR POLYGONS.

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