Chapter 8 Introductory Geometry Section 8.4 Angle Measures of Polygons

When the angle measures of polygons are discussed what is being referred to are the measures of the interior angles of the polygons. Individually these angles can have any measures, but when you add the measures of all the angles they can only be a certain number that depends on the number of sides. For example, in a triangle the measure of any angle can be any number between 0 and 180 , but if you add the interior angles together they add up to 180  m  1 + m  2 + m  3 = 180  Angle Measures of Triangles One way to see that the angles of a triangle combine to give a straight angle (i.e. measure 180  ) is to make three congruent copies of the triangle and put them together as pictured to the right. Notice the sides form a straight line. m  1 + m  2 + m  3 + m  4 = 360  m  1 + m  2 + m  3 = 180 

The way this is established formally with deductive reasoning (formal deduction van Hiele level 4) is by using the principle of alternate interior angles. Given a triangle construct a line parallel to one side going through the vertex on the opposite side.  2   4 and  3   5 (Alternate Interior Angles) m  1+m  2+m  3 = m  1+m  4+m  5 = 180  The result that the measures of interior angles of triangles is 180  form the basis for finding the interior measure of the angles of all the other polygons. This is done by breaking up the other polygons into triangles and looking at the angles of the polygons as the angles of triangles. Quadrilaterals Each of the quadrilaterals below is broken into two triangles by inserting a purple line segment in each one of them.

In the quadrilateral to the right that has been broken into two triangles we add up all the interior angles and rearrange them into two triangles m  1+ m  2+ m  3+ m  4+ m  5+ m  6 = (m  1+ m  2+ m  3)+ (m  4+ m  5+ m  6) = 180   = 360  This is another one of the patterns that exist within quadrilateral shapes is that the sum of the interior angles is always the same (like the number of diagonals). In fact, the interior angle sum of quadrilaterals is always 360 . What about other shapes? Polygons that have more sides than 4. If the polygon can always be broken apart into the same number of triangles the sum of the interior angles is always the same. Below are some examples of pentagons. 3 triangles

The interior angle sum for a pentagon can be broken apart in a similar way as a quadrilateral except you have 9 angles instead of 6. m  1+ m  2+ m  3+ m  4+ m  5 + m  6 + m  7 + m  8+ m  9 =(m  1+ m  8+ m  9)+ (m  3+ m  4 + m  5) + (m  2 + m  6+ m  7) = 180    = 540  The interior of a pentagon can always be broken into 3 triangles. A pentagon’s interior angle sum is the interior angles sum of 3 triangles which is 540 . Hexagons can always be broken into 4 triangles. The interior angle sum will be the interior angle sum of 4 triangles. Interior angle sum of a hexagon = Interior angle sum of 4 triangles = 180     = 4 · 180  = 720  4 triangles

We will use inductive reasoning to see if we can find a pattern using the entries in the table below. Name of Shape Number of Sides Number of Triangles Sum of Angles of Triangles Sum of Angles of Shape Triangle31 1·180  180  Quadrilateral42 2·180  360  Pentagon53 3·180  540  Hexagon64 4·180  720  Heptagon Octogon “n-gon” 7 5 5·180  900  n n-2 (n-2)·180  8 6 6·180  1080  The sum of the interior angles of a polygon with n sides is: (n-2)·180 

Regular Shaped Polygons A polygon is called regular if all of its sides are congruent to each other and all of its interior angles are congruent to each other. A few regular shapes you know already. A regular triangle is called an equilateral triangle. A regular quadrilateral is called a square. regular pentagon regular hexagon regular octagon Interior Angles of Regular Shaped Polygons Since each angle of a regular shaped polygon has the exact same measure we can find the measure of an angle by dividing the total angle sum by the number of angles which is also the number of sides. A regular polygon with n sides will have each of its angles measuring the following:

The formula on the previous slide cab be applied to equilateral triangles. The value of n=3 and we get the following: 60  The formula on the previous slide cab be applied to squares. The value of n=3 and we get the following: 90  How can the formula be applied to find the interior angles of the regular hexagon picture to the right? 120 