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WARM UP PROBLEM

   In    ABC, m      A = x°, m   B = 3x°, and
   m     C = (4x – 12) ° .
   Find the measures of the three angles.




ANSWER         24°, 72°, 84°
If all the angles in a triangle add up to 180⁰…

Then what about the angles in a QUADRILATERAL?
                                            Hey I see
                                               TWO
                                           triangles in
                                               there!




  2 * 180 = 3600
Or a PENTAGON?
                        Hey I see
                         THREE
                       triangles in
                           there!




3 * 180 = 540o
Or a HEXAGON?
                      Hey I see
                        FOUR
                     triangles in
                         there!




    4 * 180 = 720o
# of         Sum of
 # of                measures of
sides   triangles   interior angles

3          1          1(180)=180


4         2          2(180)=360


5         3           3(180)=540


6         4           4(180)=720

n        n-2           (n-2) • 180
If a convex polygon has n sides,
 then the sum of the measure of
       the interior angles is
      (n – 2)(180 )
If a regular convex polygon
has n sides, then the measure
of one of the interior angles is
       ( n 2)180
           n
EX. 1 USE A REGULAR 15-GON TO ANSWER THE
QUESTIONS.

 A)Find the sum of the measures of the
   interior angles. 2340


 B) Find the measure of ONE interior angle

                        156
Ex: 2 Find the value of x in the polygon
                   x

         126                    100
                                143

             130          117


126 + 130 + 117 + 143 + 100 + x = 720

                        616 + x = 720

                        x = 104
Ex: 3 The measure of each interior angle is 150°,
how many sides does the regular polygon have?
(n 2) 180
              One interior angle
    n
(n 2) 180
              150
    n
(n 2)180 150n
                               n 12
                                   A regular
180 n 360 150 n                    dodecagon
      30n 360
TWO MORE IMPORTANT TERMS
The sum of the
    measures of the            1
  exterior angles of a
 convex polygon, one
at each vertex, is 360 .

                           3
                                   2


m 1 m 2 m 3 360
The sum of the
    measures of the        1
  exterior angles of a               2
 convex polygon, one
at each vertex, is 360 .
                            4
                                3



m 1 m 2 m 3 m 4                     360
The sum of the
                               2
    measures of the
                           1
  exterior angles of a
 convex polygon, one               3
at each vertex, is 360 .
                           5
                               4



m 1 m 2 m 3 m 4 m 5 360
The measure of each exterior
angle of a regular polygon is

          360
            n
EX. 4 FIND THE MEASURE OF ONE EXTERIOR
ANGLE OF A REGULAR 20-GON.

                     sum of the exterior angles
                         number of sides
                                     
                               360
                                20

                               18
EX. 5 FIND THE MEASURE OF ONE EXTERIOR
ANGLE OF A REGULAR HEPTAGON.

 sum of the exterior angles
     number of sides
                  360
                   7
                  51.4
EX. 6 THE SUM OF THE MEASURES OF FIVE
INTERIOR ANGLES OF A HEXAGON IS 625. WHAT
IS THE MEASURE OF THE SIXTH ANGLE?




                           95

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Angle measures in polygons lesson

  • 1. WARM UP PROBLEM In ABC, m A = x°, m B = 3x°, and m C = (4x – 12) ° . Find the measures of the three angles. ANSWER 24°, 72°, 84°
  • 2.
  • 3. If all the angles in a triangle add up to 180⁰… Then what about the angles in a QUADRILATERAL? Hey I see TWO triangles in there! 2 * 180 = 3600
  • 4. Or a PENTAGON? Hey I see THREE triangles in there! 3 * 180 = 540o
  • 5. Or a HEXAGON? Hey I see FOUR triangles in there! 4 * 180 = 720o
  • 6. # of Sum of # of measures of sides triangles interior angles 3 1 1(180)=180 4 2 2(180)=360 5 3 3(180)=540 6 4 4(180)=720 n n-2 (n-2) • 180
  • 7. If a convex polygon has n sides, then the sum of the measure of the interior angles is (n – 2)(180 )
  • 8. If a regular convex polygon has n sides, then the measure of one of the interior angles is ( n 2)180 n
  • 9. EX. 1 USE A REGULAR 15-GON TO ANSWER THE QUESTIONS. A)Find the sum of the measures of the interior angles. 2340 B) Find the measure of ONE interior angle 156
  • 10. Ex: 2 Find the value of x in the polygon x 126 100 143 130 117 126 + 130 + 117 + 143 + 100 + x = 720 616 + x = 720 x = 104
  • 11. Ex: 3 The measure of each interior angle is 150°, how many sides does the regular polygon have? (n 2) 180 One interior angle n (n 2) 180 150 n (n 2)180 150n n 12 A regular 180 n 360 150 n dodecagon 30n 360
  • 13. The sum of the measures of the 1 exterior angles of a convex polygon, one at each vertex, is 360 . 3 2 m 1 m 2 m 3 360
  • 14. The sum of the measures of the 1 exterior angles of a 2 convex polygon, one at each vertex, is 360 . 4 3 m 1 m 2 m 3 m 4 360
  • 15. The sum of the 2 measures of the 1 exterior angles of a convex polygon, one 3 at each vertex, is 360 . 5 4 m 1 m 2 m 3 m 4 m 5 360
  • 16. The measure of each exterior angle of a regular polygon is 360 n
  • 17. EX. 4 FIND THE MEASURE OF ONE EXTERIOR ANGLE OF A REGULAR 20-GON. sum of the exterior angles number of sides  360 20 18
  • 18. EX. 5 FIND THE MEASURE OF ONE EXTERIOR ANGLE OF A REGULAR HEPTAGON. sum of the exterior angles number of sides 360 7 51.4
  • 19. EX. 6 THE SUM OF THE MEASURES OF FIVE INTERIOR ANGLES OF A HEXAGON IS 625. WHAT IS THE MEASURE OF THE SIXTH ANGLE? 95