The angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons.


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1 Interior Angles of Polygons The angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons. The sum of the measures of the interior angles of a triangle is 10º, so to find the sum of the measures of the interior angles of a polygon of more than three sides, see how many triangles into which the polygon can be divided with the vertices of the polygon also being the vertices of the triangles. example 1 Find the sum of the measures of the interior angles of the given hexagon. The line segments to be drawn in order to divide up the hexagon into triangles are called diagonals. By drawing three diagonals (that don t intersect), the hexagon is divided into four triangles, each with an interior angle sum of 10º. So the sum of the measures of the interior angles of the hexagon is 4 10º = 720º. The following table summarizes the sum of the measures of the interior angles of other common polygons: polygon number of sides number of triangles formed by diagonals sum of the measures of the interior angles quadrilateral º = 360º pentagon º = 40º octagon º = 100º ngon n n 2 (n 2) 10º
2 example 2 Harry measured all but one angle of a hexagon. The total degree measure for all of the angles he measured was 0º. What is the measure, in degrees, of the remaining angle? A. 92º B. 120º C. 170º D. 720º There are two ways to approach this problem. One method is to use the formula for the sum of the measures of the interior angles of a polygon, which is (n 2) 10º, and subtract 0º from the total to see what the measure of the remaining angle should be. Since there are 6 sides in a hexagon, n will equal 6: Now subtract 0º from 720º to see what the measure of the sixth interior angle is: The answer is C. The second method, if you can t remember the formula for the sum of the measures of the interior angles of a polygon, is to draw a sketch of the polygon and divide it into as many nonoverlapping triangles as you can: This was shown in example 1, and you can do this to see how many triangles you can get, as long as you remember that the sum of the measures of the interior angles of a triangle is 10º. This will also result in a sum of 720º, which you can use to find the measure of the missing angle.
3 Exterior Angles of Polygons An exterior angle of a polygon is formed by extending one side of the polygon. The sum of the measures of the exterior angles of a polygon is always 360º, no matter how many sides the polygon has or whether or not it is a regular polygon. (A regular polygon is a polygon whose sides are all congruent and whose interior angles are all congruent.) A B E C D This is true no matter how many sides a polygon has, as shown by the square and octagon below: 4º 4º 4º 4º 4º 4º 4º 4º The sum of the measures of the exterior angles of a polygon with n sides equals 360º. This means that the exterior angles of a regular polygon with n sides each has a measure of 360º divided by n. The most important aspect of an exterior angle, however, is that when it is paired up with the adjacent interior angle, the two angles make a linear pair. You should remember from Lesson 32, a linear pair is made up of two supplementary angles whose noncommon sides make a line: 1 2
4 Diagonals of Polygons Any polygon can have more diagonals than just the ones that divide a polygon into triangles. Any quadrilateral (foursided polygon) has two diagonals. Any pentagon has five diagonals. Any hexagon has nine diagonals. example 3 How many diagonals will an octagon have? There is a pattern to how many diagonals are added each time a side is added to a polygon: number of sides number of diagonals A polygon with seven sides would have 9 +, or 14, diagonals, so an octagon would have , or 20 diagonals.
5 Name For problems 1 6, find the sum of the measures of the interior angles of the polygon with the given number of sides. 1) 2) 6 3) 4) 21 ) 4 6) 7 For problems 7 10, determine the number of sides a regular polygon would have with the given value for each interior angle. [Hint: (n 2) 10º = size of interior angle n] 7) 144º ) 120º 9) 13º 10) 10º 11) The measures of four interior angles fifth interior angle. angle.
6 For problems 13 1, determine the measure of an exterior angle of a regular polygon with the given number of sides. (Remember, the sum of the measures of all exterior angles of a polygon is 360º.) 13) 14) 3 1) 16) 6 17) 4 1) 9 For problems 19 24, state the number of diagonals from each vertex that each polygon with the given number of sides has. 19) 20) 12 21) 1 22) 23) 7 24) 10 For problems 2 30, state the total number of diagonals that each polygon with the given number of sides has. 2) 26) 10 27) 11 2) 9 29) 14 30) 7
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