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Section 5-7
Diagonals and Angles of Polygons
Essential Questions


✤   How are polygons classified according to their sides?

✤   How do you find the sum of the angle measures of polygons?



✤   Where you’ll see this:

    ✤   Safety, hobbies, nature
Vocabulary
1. Polygon:


2. Side:

3. Vertex:

4. Convex:

5. Concave:

6. Regular Polygon:


7. Diagonal:
Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
     their endpoints

2. Side:

3. Vertex:

4. Convex:

5. Concave:

6. Regular Polygon:


7. Diagonal:
Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
     their endpoints

2. Side: One of the segments that makes up the polygon

3. Vertex:

4. Convex:

5. Concave:

6. Regular Polygon:


7. Diagonal:
Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
     their endpoints

2. Side: One of the segments that makes up the polygon

3. Vertex: The point where segments meet

4. Convex:

5. Concave:

6. Regular Polygon:


7. Diagonal:
Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
     their endpoints

2. Side: One of the segments that makes up the polygon

3. Vertex: The point where segments meet

4. Convex: When there are no indentations in a polygon

5. Concave:

6. Regular Polygon:


7. Diagonal:
Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
     their endpoints

2. Side: One of the segments that makes up the polygon

3. Vertex: The point where segments meet

4. Convex: When there are no indentations in a polygon

5. Concave: When there is an indentation into a polygon

6. Regular Polygon:


7. Diagonal:
Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
     their endpoints

2. Side: One of the segments that makes up the polygon

3. Vertex: The point where segments meet

4. Convex: When there are no indentations in a polygon

5. Concave: When there is an indentation into a polygon

6. Regular Polygon: A polygon where all the sides and angles are
     congruent

7. Diagonal:
Vocabulary
1. Polygon: A closed figure made by joining three or more segments at
     their endpoints

2. Side: One of the segments that makes up the polygon

3. Vertex: The point where segments meet

4. Convex: When there are no indentations in a polygon

5. Concave: When there is an indentation into a polygon

6. Regular Polygon: A polygon where all the sides and angles are
     congruent

7. Diagonal: A segment that joins two vertices but is not a side
Polygons and Their Sides

  5 sides:    6 sides:     7 sides:




  8 sides:    9 sides:     10 sides:
Polygons and Their Sides

  5 sides:    6 sides:     7 sides:
  Pentagon




  8 sides:    9 sides:     10 sides:
Polygons and Their Sides

  5 sides:    6 sides:     7 sides:
  Pentagon




  8 sides:    9 sides:     10 sides:
Polygons and Their Sides

  5 sides:     6 sides:    7 sides:
  Pentagon    Hexagon




  8 sides:     9 sides:    10 sides:
Polygons and Their Sides

  5 sides:     6 sides:    7 sides:
  Pentagon    Hexagon




  8 sides:     9 sides:    10 sides:
Polygons and Their Sides

  5 sides:     6 sides:     7 sides:
  Pentagon    Hexagon      Heptagon




  8 sides:     9 sides:     10 sides:
Polygons and Their Sides

  5 sides:     6 sides:     7 sides:
  Pentagon    Hexagon      Heptagon




  8 sides:     9 sides:     10 sides:
Polygons and Their Sides

  5 sides:     6 sides:     7 sides:
  Pentagon    Hexagon      Heptagon




  8 sides:     9 sides:     10 sides:
  Octagon
Polygons and Their Sides

  5 sides:     6 sides:     7 sides:
  Pentagon    Hexagon      Heptagon




  8 sides:     9 sides:     10 sides:
  Octagon
Polygons and Their Sides

  5 sides:     6 sides:     7 sides:
  Pentagon    Hexagon      Heptagon




  8 sides:     9 sides:     10 sides:
  Octagon     Nonagon
Polygons and Their Sides

  5 sides:     6 sides:     7 sides:
  Pentagon    Hexagon      Heptagon




  8 sides:     9 sides:     10 sides:
  Octagon     Nonagon
Polygons and Their Sides

  5 sides:     6 sides:     7 sides:
  Pentagon    Hexagon      Heptagon




  8 sides:     9 sides:     10 sides:
  Octagon     Nonagon       Decagon
Polygons and Their Sides

  5 sides:     6 sides:     7 sides:
  Pentagon    Hexagon      Heptagon




  8 sides:     9 sides:     10 sides:
  Octagon     Nonagon       Decagon
Polygons and Their Sides

  5 sides:               6 sides:    7 sides:
  Pentagon              Hexagon     Heptagon




  8 sides:               9 sides:    10 sides:
  Octagon               Nonagon      Decagon




     Anything larger:
Polygons and Their Sides

  5 sides:                  6 sides:                  7 sides:
  Pentagon                 Hexagon                  Heptagon




  8 sides:                  9 sides:                  10 sides:
  Octagon                  Nonagon                    Decagon




     Anything larger: n-gon, where n is the number of sides
Example 1
 Name each polygon by its number of sides and label as concave or
                            convex.
Example 1
 Name each polygon by its number of sides and label as concave or
                            convex.

                   Concave
Example 1
 Name each polygon by its number of sides and label as concave or
                            convex.

                   Concave
                  Pentagon
Example 1
 Name each polygon by its number of sides and label as concave or
                            convex.

                   Concave                      Convex
                  Pentagon
Example 1
 Name each polygon by its number of sides and label as concave or
                            convex.

                   Concave                      Convex
                  Pentagon                      Octagon
Example 1
 Name each polygon by its number of sides and label as concave or
                            convex.

                   Concave                      Convex
                   Pentagon                     Octagon




          Convex
Example 1
 Name each polygon by its number of sides and label as concave or
                            convex.

                   Concave                      Convex
                   Pentagon                     Octagon




          Convex
       Quadrilateral
Example 1
 Name each polygon by its number of sides and label as concave or
                            convex.

                   Concave                       Convex
                   Pentagon                      Octagon




          Convex                       Concave
       Quadrilateral
Example 1
 Name each polygon by its number of sides and label as concave or
                            convex.

                   Concave                       Convex
                   Pentagon                      Octagon




          Convex                       Concave
       Quadrilateral                  Nonagon
# of sides:       # of sides:

# of triangles:   # of triangles:

 Degrees:          Degrees:




  # of sides:       # of sides:

# of triangles:   # of triangles:

 Degrees:          Degrees:
# of sides: 3     # of sides:

# of triangles:   # of triangles:

 Degrees:          Degrees:




  # of sides:       # of sides:

# of triangles:   # of triangles:

 Degrees:          Degrees:
# of sides: 3       # of sides:

# of triangles: 1   # of triangles:

 Degrees:            Degrees:




  # of sides:         # of sides:

# of triangles:     # of triangles:

 Degrees:            Degrees:
# of sides: 3       # of sides:

# of triangles: 1   # of triangles:

 Degrees: 180°       Degrees:




  # of sides:         # of sides:

# of triangles:     # of triangles:

 Degrees:            Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles:

 Degrees: 180°       Degrees:




  # of sides:         # of sides:

# of triangles:     # of triangles:

 Degrees:            Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles:

 Degrees: 180°       Degrees:




  # of sides:         # of sides:

# of triangles:     # of triangles:

 Degrees:            Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees:




  # of sides:         # of sides:

# of triangles:     # of triangles:

 Degrees:            Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




  # of sides:         # of sides:

# of triangles:     # of triangles:

 Degrees:            Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




  # of sides: 5       # of sides:

# of triangles:     # of triangles:

 Degrees:            Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




  # of sides: 5       # of sides:

# of triangles:     # of triangles:

 Degrees:            Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




  # of sides: 5       # of sides:

# of triangles:     # of triangles:

 Degrees:            Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




 # of sides: 5        # of sides:

# of triangles: 3   # of triangles:

 Degrees:            Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




 # of sides: 5        # of sides:

# of triangles: 3   # of triangles:

 Degrees: 540°       Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




 # of sides: 5        # of sides: 6

# of triangles: 3   # of triangles:

 Degrees: 540°       Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




 # of sides: 5        # of sides: 6

# of triangles: 3   # of triangles:

 Degrees: 540°       Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




 # of sides: 5        # of sides: 6

# of triangles: 3   # of triangles:

 Degrees: 540°       Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




 # of sides: 5        # of sides: 6

# of triangles: 3   # of triangles:

 Degrees: 540°       Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




 # of sides: 5       # of sides: 6

# of triangles: 3   # of triangles: 4

 Degrees: 540°       Degrees:
# of sides: 3       # of sides: 4

# of triangles: 1   # of triangles: 2

 Degrees: 180°       Degrees: 360°




 # of sides: 5       # of sides: 6

# of triangles: 3   # of triangles: 4

 Degrees: 540°       Degrees: 720°
Angle Sum of a Polygon:




Angle Measure of a Regular Polygon:
Angle Sum of a Polygon: The sum of the interior angles of a polygon
   with n sides is given by the formula




Angle Measure of a Regular Polygon:
Angle Sum of a Polygon: The sum of the interior angles of a polygon
   with n sides is given by the formula




Angle Measure of a Regular Polygon: The measure of each interior
  angle of a regular polygon with n sides is given by the formula
       (n − 2)180°
    S=
            n
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A


 F                       B




 E                       C


             D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
 F                         B
                     10x




 E                         C


             D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
 F                           B
                      10x



                    3x + 8
 E                           C
           3x + 8

             D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
 F                               B
                          10x



     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
 F                               B
     8x - 12              10x



     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                        S = (n − 2)180°
 F                               B
     8x - 12              10x



     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                        S = (n − 2)180°
 F                               B
     8x - 12              10x           S = (6 − 2)180°


     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                        S = (n − 2)180°
 F                               B
     8x - 12              10x           S = (6 − 2)180°
                                          S = (4)180°
     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                        S = (n − 2)180°
 F                               B
     8x - 12              10x           S = (6 − 2)180°
                                          S = (4)180°
     7x - 22            3x + 8             S = 720°
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                              S = (n − 2)180°
 F                               B
     8x - 12              10x                 S = (6 − 2)180°
                                                S = (4)180°
     7x - 22            3x + 8                   S = 720°
 E                               C
               3x + 8                The sum of all of the angles is 720°

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
 F                               B
     8x - 12              10x



     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x     2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
 F                               B
     8x - 12              10x



     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x     2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
                                     20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
 F                               B
     8x - 12              10x



     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x     2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
                                     20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
 F                               B
     8x - 12              10x
                                                 41x − 18 = 720°


     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x     2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
                                     20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
 F                               B
     8x - 12              10x
                                                 41x − 18 = 720°
                                                     +18 +18

     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x     2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
                                     20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
 F                               B
     8x - 12              10x
                                                 41x − 18 = 720°
                                                     +18 +18
                                                     41x = 738
     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x     2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
                                     20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
 F                               B
     8x - 12              10x
                                                 41x − 18 = 720°
                                                     +18 +18
                                                     41x = 738
                                                      41     41
     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x     2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720°
                                     20x + 6x + 16 + 7x − 22 + 8x − 12 = 720°
 F                               B
     8x - 12              10x
                                                 41x − 18 = 720°
                                                     +18 +18
                                                     41x = 738
                                                      41     41
     7x - 22            3x + 8
 E                               C                      x = 18
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                              x = 18
 F                               B
     8x - 12              10x



     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                                x = 18
 F                               B
     8x - 12              10x        10(18) =



     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                               x = 18
 F                               B
     8x - 12              10x        10(18) = 180°



     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                              x = 18
 F                               B
     8x - 12              10x        10(18) = 180° = m∠A = m∠B



     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                                   x = 18
 F                               B
     8x - 12              10x        10(18) = 180° = m∠A = m∠B
                                     3(18) + 8 =

     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                               x = 18
 F                               B
     8x - 12              10x        10(18) = 180° = m∠A = m∠B
                                     3(18) + 8 = 62°

     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                              x = 18
 F                               B
     8x - 12              10x        10(18) = 180° = m∠A = m∠B
                                     3(18) + 8 = 62° = m∠C = m∠D

     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                               x = 18
 F                               B
     8x - 12              10x        10(18) = 180° = m∠A = m∠B
                                     3(18) + 8 = 62° = m∠C = m∠D

                                      7(18) - 22 =
     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                               x = 18
 F                               B
     8x - 12              10x        10(18) = 180° = m∠A = m∠B
                                     3(18) + 8 = 62° = m∠C = m∠D

                                      7(18) - 22 = 104°
     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                              x = 18
 F                               B
     8x - 12              10x        10(18) = 180° = m∠A = m∠B
                                     3(18) + 8 = 62° = m∠C = m∠D

                                      7(18) - 22 = 104° = m∠E
     7x - 22            3x + 8
 E                               C
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                               x = 18
 F                               B
     8x - 12              10x        10(18) = 180° = m∠A = m∠B
                                     3(18) + 8 = 62° = m∠C = m∠D

                                      7(18) - 22 = 104° = m∠E
     7x - 22            3x + 8
 E                               C
                                      8(18) - 12 =
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                               x = 18
 F                               B
     8x - 12              10x        10(18) = 180° = m∠A = m∠B
                                     3(18) + 8 = 62° = m∠C = m∠D

                                      7(18) - 22 = 104° = m∠E
     7x - 22            3x + 8
 E                               C
                                      8(18) - 12 = 132°
               3x + 8

                 D
Example 2
  In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8,
 m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of
  the angles of the hexagon, then find the measure of each angle.
             A
            10x
                                              x = 18
 F                               B
     8x - 12              10x        10(18) = 180° = m∠A = m∠B
                                     3(18) + 8 = 62° = m∠C = m∠D

                                      7(18) - 22 = 104° = m∠E
     7x - 22            3x + 8
 E                               C
                                      8(18) - 12 = 132° = m∠F
               3x + 8

                 D
Example 3
    Find the measure of each angle of a regular 14-gon.
Example 3
    Find the measure of each angle of a regular 14-gon.


         (n − 2)180°
      S=
              n
Example 3
    Find the measure of each angle of a regular 14-gon.


         (n − 2)180°
      S=
              n
        (14 − 2)180°
     S=
             14
Example 3
    Find the measure of each angle of a regular 14-gon.


         (n − 2)180°
      S=
              n
        (14 − 2)180°
     S=
             14
          (12)180°
       S=
             14
Example 3
    Find the measure of each angle of a regular 14-gon.


         (n − 2)180°
      S=
              n
        (14 − 2)180°
     S=
             14
          (12)180°
       S=
             14
           2160°
        S=
            14
Example 3
    Find the measure of each angle of a regular 14-gon.


         (n − 2)180°
      S=
              n
        (14 − 2)180°
     S=
             14
                                        S = 154 2 7 °
          (12)180°
       S=
             14
           2160°
        S=
            14
Homework
Homework



                           p. 224 #1-33 odd




“Liberty without learning is always in peril; learning without liberty is
                  always in vain.” - John F. Kennedy

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Integrated Math 2 Section 5-7

  • 1. Section 5-7 Diagonals and Angles of Polygons
  • 2. Essential Questions ✤ How are polygons classified according to their sides? ✤ How do you find the sum of the angle measures of polygons? ✤ Where you’ll see this: ✤ Safety, hobbies, nature
  • 3. Vocabulary 1. Polygon: 2. Side: 3. Vertex: 4. Convex: 5. Concave: 6. Regular Polygon: 7. Diagonal:
  • 4. Vocabulary 1. Polygon: A closed figure made by joining three or more segments at their endpoints 2. Side: 3. Vertex: 4. Convex: 5. Concave: 6. Regular Polygon: 7. Diagonal:
  • 5. Vocabulary 1. Polygon: A closed figure made by joining three or more segments at their endpoints 2. Side: One of the segments that makes up the polygon 3. Vertex: 4. Convex: 5. Concave: 6. Regular Polygon: 7. Diagonal:
  • 6. Vocabulary 1. Polygon: A closed figure made by joining three or more segments at their endpoints 2. Side: One of the segments that makes up the polygon 3. Vertex: The point where segments meet 4. Convex: 5. Concave: 6. Regular Polygon: 7. Diagonal:
  • 7. Vocabulary 1. Polygon: A closed figure made by joining three or more segments at their endpoints 2. Side: One of the segments that makes up the polygon 3. Vertex: The point where segments meet 4. Convex: When there are no indentations in a polygon 5. Concave: 6. Regular Polygon: 7. Diagonal:
  • 8. Vocabulary 1. Polygon: A closed figure made by joining three or more segments at their endpoints 2. Side: One of the segments that makes up the polygon 3. Vertex: The point where segments meet 4. Convex: When there are no indentations in a polygon 5. Concave: When there is an indentation into a polygon 6. Regular Polygon: 7. Diagonal:
  • 9. Vocabulary 1. Polygon: A closed figure made by joining three or more segments at their endpoints 2. Side: One of the segments that makes up the polygon 3. Vertex: The point where segments meet 4. Convex: When there are no indentations in a polygon 5. Concave: When there is an indentation into a polygon 6. Regular Polygon: A polygon where all the sides and angles are congruent 7. Diagonal:
  • 10. Vocabulary 1. Polygon: A closed figure made by joining three or more segments at their endpoints 2. Side: One of the segments that makes up the polygon 3. Vertex: The point where segments meet 4. Convex: When there are no indentations in a polygon 5. Concave: When there is an indentation into a polygon 6. Regular Polygon: A polygon where all the sides and angles are congruent 7. Diagonal: A segment that joins two vertices but is not a side
  • 11. Polygons and Their Sides 5 sides: 6 sides: 7 sides: 8 sides: 9 sides: 10 sides:
  • 12. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon 8 sides: 9 sides: 10 sides:
  • 13. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon 8 sides: 9 sides: 10 sides:
  • 14. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon 8 sides: 9 sides: 10 sides:
  • 15. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon 8 sides: 9 sides: 10 sides:
  • 16. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon Heptagon 8 sides: 9 sides: 10 sides:
  • 17. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon Heptagon 8 sides: 9 sides: 10 sides:
  • 18. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon Heptagon 8 sides: 9 sides: 10 sides: Octagon
  • 19. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon Heptagon 8 sides: 9 sides: 10 sides: Octagon
  • 20. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon Heptagon 8 sides: 9 sides: 10 sides: Octagon Nonagon
  • 21. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon Heptagon 8 sides: 9 sides: 10 sides: Octagon Nonagon
  • 22. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon Heptagon 8 sides: 9 sides: 10 sides: Octagon Nonagon Decagon
  • 23. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon Heptagon 8 sides: 9 sides: 10 sides: Octagon Nonagon Decagon
  • 24. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon Heptagon 8 sides: 9 sides: 10 sides: Octagon Nonagon Decagon Anything larger:
  • 25. Polygons and Their Sides 5 sides: 6 sides: 7 sides: Pentagon Hexagon Heptagon 8 sides: 9 sides: 10 sides: Octagon Nonagon Decagon Anything larger: n-gon, where n is the number of sides
  • 26. Example 1 Name each polygon by its number of sides and label as concave or convex.
  • 27. Example 1 Name each polygon by its number of sides and label as concave or convex. Concave
  • 28. Example 1 Name each polygon by its number of sides and label as concave or convex. Concave Pentagon
  • 29. Example 1 Name each polygon by its number of sides and label as concave or convex. Concave Convex Pentagon
  • 30. Example 1 Name each polygon by its number of sides and label as concave or convex. Concave Convex Pentagon Octagon
  • 31. Example 1 Name each polygon by its number of sides and label as concave or convex. Concave Convex Pentagon Octagon Convex
  • 32. Example 1 Name each polygon by its number of sides and label as concave or convex. Concave Convex Pentagon Octagon Convex Quadrilateral
  • 33. Example 1 Name each polygon by its number of sides and label as concave or convex. Concave Convex Pentagon Octagon Convex Concave Quadrilateral
  • 34. Example 1 Name each polygon by its number of sides and label as concave or convex. Concave Convex Pentagon Octagon Convex Concave Quadrilateral Nonagon
  • 35. # of sides: # of sides: # of triangles: # of triangles: Degrees: Degrees: # of sides: # of sides: # of triangles: # of triangles: Degrees: Degrees:
  • 36. # of sides: 3 # of sides: # of triangles: # of triangles: Degrees: Degrees: # of sides: # of sides: # of triangles: # of triangles: Degrees: Degrees:
  • 37. # of sides: 3 # of sides: # of triangles: 1 # of triangles: Degrees: Degrees: # of sides: # of sides: # of triangles: # of triangles: Degrees: Degrees:
  • 38. # of sides: 3 # of sides: # of triangles: 1 # of triangles: Degrees: 180° Degrees: # of sides: # of sides: # of triangles: # of triangles: Degrees: Degrees:
  • 39. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: Degrees: 180° Degrees: # of sides: # of sides: # of triangles: # of triangles: Degrees: Degrees:
  • 40. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: Degrees: 180° Degrees: # of sides: # of sides: # of triangles: # of triangles: Degrees: Degrees:
  • 41. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: # of sides: # of sides: # of triangles: # of triangles: Degrees: Degrees:
  • 42. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: # of sides: # of triangles: # of triangles: Degrees: Degrees:
  • 43. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: 5 # of sides: # of triangles: # of triangles: Degrees: Degrees:
  • 44. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: 5 # of sides: # of triangles: # of triangles: Degrees: Degrees:
  • 45. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: 5 # of sides: # of triangles: # of triangles: Degrees: Degrees:
  • 46. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: 5 # of sides: # of triangles: 3 # of triangles: Degrees: Degrees:
  • 47. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: 5 # of sides: # of triangles: 3 # of triangles: Degrees: 540° Degrees:
  • 48. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: 5 # of sides: 6 # of triangles: 3 # of triangles: Degrees: 540° Degrees:
  • 49. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: 5 # of sides: 6 # of triangles: 3 # of triangles: Degrees: 540° Degrees:
  • 50. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: 5 # of sides: 6 # of triangles: 3 # of triangles: Degrees: 540° Degrees:
  • 51. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: 5 # of sides: 6 # of triangles: 3 # of triangles: Degrees: 540° Degrees:
  • 52. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: 5 # of sides: 6 # of triangles: 3 # of triangles: 4 Degrees: 540° Degrees:
  • 53. # of sides: 3 # of sides: 4 # of triangles: 1 # of triangles: 2 Degrees: 180° Degrees: 360° # of sides: 5 # of sides: 6 # of triangles: 3 # of triangles: 4 Degrees: 540° Degrees: 720°
  • 54. Angle Sum of a Polygon: Angle Measure of a Regular Polygon:
  • 55. Angle Sum of a Polygon: The sum of the interior angles of a polygon with n sides is given by the formula Angle Measure of a Regular Polygon:
  • 56. Angle Sum of a Polygon: The sum of the interior angles of a polygon with n sides is given by the formula Angle Measure of a Regular Polygon: The measure of each interior angle of a regular polygon with n sides is given by the formula (n − 2)180° S= n
  • 57. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle.
  • 58. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A F B E C D
  • 59. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x F B 10x E C D
  • 60. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x F B 10x 3x + 8 E C 3x + 8 D
  • 61. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x F B 10x 7x - 22 3x + 8 E C 3x + 8 D
  • 62. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x F B 8x - 12 10x 7x - 22 3x + 8 E C 3x + 8 D
  • 63. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x S = (n − 2)180° F B 8x - 12 10x 7x - 22 3x + 8 E C 3x + 8 D
  • 64. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x S = (n − 2)180° F B 8x - 12 10x S = (6 − 2)180° 7x - 22 3x + 8 E C 3x + 8 D
  • 65. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x S = (n − 2)180° F B 8x - 12 10x S = (6 − 2)180° S = (4)180° 7x - 22 3x + 8 E C 3x + 8 D
  • 66. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x S = (n − 2)180° F B 8x - 12 10x S = (6 − 2)180° S = (4)180° 7x - 22 3x + 8 S = 720° E C 3x + 8 D
  • 67. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x S = (n − 2)180° F B 8x - 12 10x S = (6 − 2)180° S = (4)180° 7x - 22 3x + 8 S = 720° E C 3x + 8 The sum of all of the angles is 720° D
  • 68. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x F B 8x - 12 10x 7x - 22 3x + 8 E C 3x + 8 D
  • 69. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720° F B 8x - 12 10x 7x - 22 3x + 8 E C 3x + 8 D
  • 70. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720° 20x + 6x + 16 + 7x − 22 + 8x − 12 = 720° F B 8x - 12 10x 7x - 22 3x + 8 E C 3x + 8 D
  • 71. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720° 20x + 6x + 16 + 7x − 22 + 8x − 12 = 720° F B 8x - 12 10x 41x − 18 = 720° 7x - 22 3x + 8 E C 3x + 8 D
  • 72. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720° 20x + 6x + 16 + 7x − 22 + 8x − 12 = 720° F B 8x - 12 10x 41x − 18 = 720° +18 +18 7x - 22 3x + 8 E C 3x + 8 D
  • 73. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720° 20x + 6x + 16 + 7x − 22 + 8x − 12 = 720° F B 8x - 12 10x 41x − 18 = 720° +18 +18 41x = 738 7x - 22 3x + 8 E C 3x + 8 D
  • 74. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720° 20x + 6x + 16 + 7x − 22 + 8x − 12 = 720° F B 8x - 12 10x 41x − 18 = 720° +18 +18 41x = 738 41 41 7x - 22 3x + 8 E C 3x + 8 D
  • 75. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x 2(10x) + 2(3x + 8) + 7x − 22 + 8x − 12 = 720° 20x + 6x + 16 + 7x − 22 + 8x − 12 = 720° F B 8x - 12 10x 41x − 18 = 720° +18 +18 41x = 738 41 41 7x - 22 3x + 8 E C x = 18 3x + 8 D
  • 76. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 7x - 22 3x + 8 E C 3x + 8 D
  • 77. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 7x - 22 3x + 8 E C 3x + 8 D
  • 78. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 180° 7x - 22 3x + 8 E C 3x + 8 D
  • 79. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 180° = m∠A = m∠B 7x - 22 3x + 8 E C 3x + 8 D
  • 80. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 180° = m∠A = m∠B 3(18) + 8 = 7x - 22 3x + 8 E C 3x + 8 D
  • 81. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 180° = m∠A = m∠B 3(18) + 8 = 62° 7x - 22 3x + 8 E C 3x + 8 D
  • 82. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 180° = m∠A = m∠B 3(18) + 8 = 62° = m∠C = m∠D 7x - 22 3x + 8 E C 3x + 8 D
  • 83. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 180° = m∠A = m∠B 3(18) + 8 = 62° = m∠C = m∠D 7(18) - 22 = 7x - 22 3x + 8 E C 3x + 8 D
  • 84. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 180° = m∠A = m∠B 3(18) + 8 = 62° = m∠C = m∠D 7(18) - 22 = 104° 7x - 22 3x + 8 E C 3x + 8 D
  • 85. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 180° = m∠A = m∠B 3(18) + 8 = 62° = m∠C = m∠D 7(18) - 22 = 104° = m∠E 7x - 22 3x + 8 E C 3x + 8 D
  • 86. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 180° = m∠A = m∠B 3(18) + 8 = 62° = m∠C = m∠D 7(18) - 22 = 104° = m∠E 7x - 22 3x + 8 E C 8(18) - 12 = 3x + 8 D
  • 87. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 180° = m∠A = m∠B 3(18) + 8 = 62° = m∠C = m∠D 7(18) - 22 = 104° = m∠E 7x - 22 3x + 8 E C 8(18) - 12 = 132° 3x + 8 D
  • 88. Example 2 In hexagon ABCDEF, m∠A = m∠B = 10x, m∠C = m∠D = 3x + 8, m∠E = 7x - 22, and m∠F = 8x - 12. Find the sum of the measures of the angles of the hexagon, then find the measure of each angle. A 10x x = 18 F B 8x - 12 10x 10(18) = 180° = m∠A = m∠B 3(18) + 8 = 62° = m∠C = m∠D 7(18) - 22 = 104° = m∠E 7x - 22 3x + 8 E C 8(18) - 12 = 132° = m∠F 3x + 8 D
  • 89. Example 3 Find the measure of each angle of a regular 14-gon.
  • 90. Example 3 Find the measure of each angle of a regular 14-gon. (n − 2)180° S= n
  • 91. Example 3 Find the measure of each angle of a regular 14-gon. (n − 2)180° S= n (14 − 2)180° S= 14
  • 92. Example 3 Find the measure of each angle of a regular 14-gon. (n − 2)180° S= n (14 − 2)180° S= 14 (12)180° S= 14
  • 93. Example 3 Find the measure of each angle of a regular 14-gon. (n − 2)180° S= n (14 − 2)180° S= 14 (12)180° S= 14 2160° S= 14
  • 94. Example 3 Find the measure of each angle of a regular 14-gon. (n − 2)180° S= n (14 − 2)180° S= 14 S = 154 2 7 ° (12)180° S= 14 2160° S= 14
  • 96. Homework p. 224 #1-33 odd “Liberty without learning is always in peril; learning without liberty is always in vain.” - John F. Kennedy

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