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19.) A.) use the idea of walking and turning around a shape to determine the sum of the exterior angles of the quadrilateral and figured 10.93. In other words, determine e + f + g + h. Measure with a protractor m to check that your formula is correct for this quadrilateral. B.) Will there be a similar formula for the sum of the exterior angles of Pentagon ‘s, hexagons, heptagon ‘s, octagons, and so on? Explain.C.) using your for your formula for the sum of the exterior angles of a quadrilateral, reduce the sum of the interior angles of the quadrilateral.in other words, find a+b+c+d, as pictured in figure 10.93. Explain your reasoning. Measure with a protractor to verify that your formula is correct for this quadrilateral.D.) based on your work, what formula would you expect to be true for the sum of the interior angles of a pentagon? what about for a hexagon? What about for a polygon with 10 sides? Explain briefly

19 A Use The Idea Of Walking And Turning Around A Shape To Determine The Sum Of The Exterior Angles Of The Quadrilateral And Figured 1093 In Other Words Determi class=

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A) To find the external angles of the quadrilateral we will walk around the shape measuring every turn we take. We will start on the A point, rotate clock-wise and move in the direction of point "B", when we get there we will rotate clock-wise again and walk to the direction of point "C". When we do get to the point C we will notice that we rotated 180 degrees in relation to the initial position we had in point A. Moving forwars we will now rotate clockwise and go to the poind D, rotate clock-wise again when we get there, performing all the rotations needed. We will notice that we have the same orientation from the beginning, this means that we rotated 360 degrees. In other words the sum of the external angles of the quadrilateral is 360 degrees.

B) Yes, any regular polygon will have the sum of its external angles equal to 360 degrees.

C) The internal and external angles are suplementary. This means that the sum of these angles must be equal to 180 degrees, therefore:

[tex]\begin{gathered} external\text{ = 180-internal} \\ e+f+g+h=360 \\ (180-a)+(180-b)+(180-c)+(180-d)=360 \\ a+b+c+d=180+180+180+180-360 \\ a+b+c+d=4\cdot180-2\cdot180 \\ a+b+c+d=(4-2)\cdot180 \\ a+b+c+d=2\cdot180=360 \end{gathered}[/tex]

Since each external angle is the same as "180 degrees" minus the internal angle that is close to it we can represent the sum of the external angles as 360 degrees and use the mentioned relation to convert them into internal angles. If we isolate them as a sum we will find the value of the sum of the internal angles.

D) If we look at the fith line from the solution above we will notice that the sum of internal angles is represented by "(4-2)*180", the polygon had "4" sides. This means that for one that is 5 sides we should expect that it would be "(5-2)*180" and so on. So the formula is:

[tex]\text{internal = (n-2)}\cdot180[/tex]

Where "n" is the number of sides of the polygon.

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