Geometry                                                                     Name:  _______________________

Lesson 11.1:  Angle Measures in Polygons                         Date:  __________________

 

Essential Question :  How is the measure of an interior angle or an exterior angle of a polygon found?

Concepts that you need to remember for today’s lesson:

1.  What is the sum of the measures of the angles in a triangle?  __________

 

2.  What is inductive reasoning?         __________________________________________

 

________________________________________________________________________

 

3.  What is a regular polygon? __________________________________________

 

4.  What is the sum of the degrees around a single point?                       __________

 

Complete the following chart:          

Diagram of                   Number            Number of                    Sum of the measures

a polygon:                     of sides:            triangles insideof the interior angles:

 

  


 

                                    3                      1                                  180° x 1 = __________

 

 


 

                                    4                      2                                  180° x 2 = __________

 

  


 

                                    ____                ____                            ___________________

 

 

 


 

                                    ____                ____                            ___________________

 

 

 


 

                                    ____                ____                            ___________________

 

  

 


 

                                    ____                ____                            ___________________

 

Notice that there is a pattern in the relationship that exists between the number of sides a shape has and the number of triangles that can be drawn inside it.  If a polygon has n sides, how many triangles can be drawn inside the polygon?  ______________

  


 

If a polygon has n sides, what is the sum of the measures of its interior angles?

(This is an important formula!!!)


 

What is the formula for finding the sum of the measures of the interior angles in a polygon with n sides?

 

 

Example 1:  Find the sum of the measures of the interior angles of ...

a.  ... a hexagon                        b. ... a nonagon (9-sided)          c.  ... a 14 – gon

 

 

 

 

Since the angles in a regular polygon are congruent, the measure of one interior angle of a regular polygon can be found by dividing the sum of the measures of the interior angles by the number of angles. 

The formula for the measure of each interior angle of a regular polygon

(Another important formula!!!)

 

Example 2:  Find the measure of one interior angle for ...

a.  ... a regular pentagon            b.  ... a regular octagon c.  … a regular 12-gon

 

 

 

 

 

 

 

 

 

Example 3:  The measure of each interior angle of a regular polygon is 165°.  How many sides does the polygon have? 

 

 

 

 

 

 

 

 

 

 


 

 

Consider the following diagram of a convex polygon and with its exterior          angles shaded. 

If we were to cut out the exterior angles we could slide them together at their vertices.


 

                                                           

  

 

 

 

 

 

 

 


 

Notice that when they are brought together, the 3 exterior angles fit together perfectly all the way around a point.  How many degrees are there all the way around a point?  ______ 

The diagram can be extended to polygons with different numbers of sides.  

The formula for the sum of the measures of the exterior angles of a polygon:   

(Again, an important formula!!!)

Example 4:  Find the sum of the measures of the exterior angles for ...

a.  ... a quadrilateral                  b.  ... a pentagon                       c.  ... a n-gon

 

 

Since the angles in a regular polygon are all congruent, the exterior angles are also all congruent.  This means that the measure of one exterior angle of a regular polygon can be found by dividing the sum of the measures of the exterior angles by the number of exterior angles. 

The formula for the measure of each exterior angle of a regular polygon

(Yet another important formula!!!)

 

Example 5:  Find the measure of each exterior angle for a regular polygon that is ...

a. ... a quadrilateral                   b.  ... a pentagon                       c. ... a n-gon

 

 

 

Example 6:  Find the number of sides the regular polygon has if the measure of one exterior angle is ...

a.  ... 60˚                                                          b.  ... 30˚