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What is a Polygon?  A polygon has at least three or more sides to form a convex or irregular shape. -Therefore the interior angles of a polygon involve the angles INSIDE the polygon!

What are Interior Angles?  Interior angles are the angles inside any shape  In this situation the interior angles are applied to polygons  The sum of the interior angles depends on the number of sides of the polygon

What is the Polygon Interior Angles Sum Theorem?  The PIAST is the sum of the interior angles dependent on the number of sides of the polygon.  Use this formula: (n-2)180  Follow the steps below: “n” stands for the number of sides of the polygon Once you’ve counted the number of sides subtract by 2 Then, multiply 180 by (n-2)

Tips & Additional Info.  A nifty tip is to just remember the formula: (n-2)180  If you forget the multiplication of 180, just know that the smallest number of any interior angle sum is 180 (a triangle)

Triangle Sum Theorem  Aside from the PIAST, the Triangle Sum Theorem can help when remembering sum angles and is a relatable theorem  So…What is it them? All interior angles of a triangle will ALWAYS equal 180 Equilateral, isosceles, or scalene all angles will add to 180  How is this Useful for the PIAST? This theorem is handy because if you forget the formula for the PIAST then you can remember the Triangle Sum Theorem and know that 180 is the smallest angle sum for any shape!

How does this Theorem apply?  The Polygon Interior Angles Sum Theorem is useful when trying to find a missing interior angle.  Once you find the amount of sides subtract by 2 and then multiply by 180, (n-2)180, you then know the total amount of angles the polygon will have.

Example No. 1  Figure out the following problem: If Zheng He has 2 triangles, 3 octagons, and 7 12-gons, he wants to combine all the shapes into one unison figure. Show your work giving the exact number of interior angles. In two forms, separate shapes and then together as a whole.  Explanation: When you use the PIAST you will be able to figure out the interior angles for the triangles, octagons, and 12-gons, by using the formula (n-2)180. Then once you have figured out the sum angles for all shapes you then use addition to find the interior angles for the entire figure. 12-gons (7): 12-2=10x180= x7= Octagons (3): 8-2=6x180= x3= 3240 Triangles (2): 3-2=1x180= x2= 360 The total number of angles is: Triangles: Octagons: gons: 12600

Example No.2  Below is a figure displaying interior angles. Find the value of x. What do we know: This a hexagon-6 sided polygon Therefore use (n-2)180 (6-2)180= 720 Actual Work: x=720 Combine like terms 600+x= X=120 Explanation: When using the Polygon Interior Angles Sum Theorem, you will be able to find the total amount of sides of the polygon. Thus you will be able to find the total angle sum, and find the value of x.

5 Practice Problems  1) If n=7 what is the sum of the interior angles?  2) If n=54 what is the sum of the interior angles?  3) What are the interior angles of congruent decagons?  4) Find the interior angles of a congruent 17-gon  5) If a hexagon has the following angles: 100, 50, 75, 115, 185, and x what is the value of x?

Answers to Practice Problems  1) 900  2) 9360  3) 1440  4) 2700  5) x= 195