1. Angles of Polygons
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Draw a quadrilateral, divide it into two triangles, and use the Triangle Sum Theorem. Do the same with a polygon with more sides. Look for a relation between the number of sides and the sum of the measures of the interior angles.
(n-2)180^(∘), where n is the number of sides of the polygon.
By the Triangle Sum Theorem, we know that the sum of the interior angles of a triangle is equal to 180^(∘).
To find a general result, we will find the required sum for a quadrilateral, pentagon, and hexagon. This will give us a relation between the number of sides and the sum of the measures of the interior angles of a polygon.
Let's consider a general quadrilateral and let's draw one diagonal which divides it into two triangles.
Let's draw a pentagon and let's draw one diagonal which divides the pentagon into a quadrilateral and a triangle.
By using the Triangle Sum Theorem, and the result for quadrilaterals we found above, we can write the following two equations. m ∠1 + m ∠2 + m ∠3 + m ∠4 = 360^(∘) & (I) m ∠5 + m ∠6 + m ∠7 = 180^(∘) & (II) Next, we can add the two equations above and use the Angle Addition Postulate. m∠A + m∠B + m∠C + m∠D + m∠E = 540^(∘) Thus, the sum of the measures of the interior angles of a pentagon is equal to 540^(∘).
Below we draw a general hexagon and also one diagonal such that it divides the hexagon into a pentagon and a triangle.
From what we've found so far, we know that the sum of the measures of the interior angles of the pentagon is 540^(∘) and the sum of the measures of the interior angles of the triangle is 180^(∘). Sum of angles ABDEF = 540^(∘) & (I) Sum of angles BCD = 180^(∘) & (II) In consequence, if we add the two equations above we will get that the sum of the measures of the interior angles of a hexagon is equal to 720^(∘).
Let's make a table with the four results we know so far.
| Polygon | Sides | Sum of Interior Angles |
|---|---|---|
| Triangle | 3 | 180^(∘) |
| Quadrilateral | 4 | 360^(∘) |
| Pentagon | 5 | 540^(∘) |
| Hexagon | 6 | 720^(∘) |
Notice that each number in the third column is a multiple of 180. Using this, we will rewrite these numbers in terms of the number of sides of the polygon.
| Polygon | Sides | Sum of Interior Angles |
|---|---|---|
| Triangle | 3 | 180^(∘) =( 3-2)180^(∘) |
| Quadrilateral | 4 | 2* 180^(∘) = ( 4-2)180^(∘) |
| Pentagon | 5 | 3* 180^(∘) = ( 5-2)180^(∘) |
| Hexagon | 6 | 4* 180^(∘) = ( 6-2)180^(∘) |
Can you see the pattern? From the above, we conclude that the sum of the measures of the interior angles of a polygon with n sides is equal to (n-2)180^(∘).
| Polygon | Sides | Sum of Interior Angles |
|---|---|---|
| Polygon | n | ( n-2)180^(∘) |