all right 8.1 fine angle measures and polygons I think most you guys already know what a diagonal is a diagonal of a polygon is a segment that joins two non-consecutive vertices remember consecutive means they're right next to each other non consecutive means the vertices are not right next to each other okay so for example in this quadrilateral this would be a diagonal okay in this Pentagon this would be a diagonal all right because this versus this vertex and this vertex are non-consecutive all right theorem eight point one polygon interior angles the sum of the measures of the interior angles of a convex no this is only for convex and gonns is n minus two times 180 so if you were to add up all the angles inside any polygon the sum would be the number of sides minus two times 180 and we'll do an example of that in a sec all right a corollary the sum of the measures of the interior angles of a quadrilateral metal quadrilateral has four sides is 360 any quadrilateral if you add up all four angles you get 360 degrees all right so for example find the sum of the measures of the interior angles of a convex hexagon okay now remember a hexagon has six sides so we're going to use the polygon interior angle stamp and minus two times 180 in this case our n is 6 6 minus 2 times 180 I plugged in 6 for n because it has six sides 6 minus 2 is 4 4 times 180 7:20 so the sum of the measures of the interior angles of a hexagon is 720 degrees all right let's go on to page 2 okay the sum of the measures of the interior angles of a convex polygon is twelve hundred sixty degrees classify the polygon by the number of sides all right so now we have to work backwards use the polygon interior angles theorem to write an equation involving the number of sides and then solve the equation to find the number of sides so we know our equation is n minus two times 180 but in this case we don't know what n is we need to solve for n but we know that whatever n equals we're going to end up with twelve hundred and sixty degrees okay I'm going to divide both sides by 180 so if you divide this side by 180 other side by 180 you get n minus two equals let's pull out the calculator [Music] yeah no way 1260 divided by 187 okay now I'm going to add two to each side and equals nine so the polygon has nine sides if you remember a nine sided figure is called a nonagon all right okay example three find the value of x in the diagram shown the polygon is a quadrilateral use the corollary of to the polygon interior angles theorem to write an equation of all the next okay so if you guys remember we just learned that a quadrilateral all the angles are gonna add up to 360 now if you were to plug in n minus 2 times 180 if n is 4 2 times 180 is 360 you would get the same thing this is just kind of a shortcut if you just remember any quadrilateral they add up to 360 you're good so if I all four of these angles I'm gonna get 360 so X degrees plus 71 degrees plus 135 degrees plus 112 degrees equals 360 now I just need to solve for X if I add all three of these together let's see here 135 112 71 and that would be 8 11 318 so X plus 3 18 equals 360 I'm gonna subtract 3 18 from both sides 360 minus 3 18 need to borrow 42 so x equals 42 in this case 42 degrees all right I'll let you guys do the checkpoint let's go onto page 3 I'll let you guys do these two checkpoints also okay theorem eight point two polygons exterior angles theorem the sum of the measures of the exterior angles of a convex convex note that it doesn't work for concave polygons convex polygons one angle at each vertex is 360 degrees this is actually a I think a much easier theorem to use and the other one because no matter how many sides a polygon has there all the exterior angles are going to add up to 360 degrees with the interior angles you're going to get a different some depending on how many sides the polygon has but the exterior angles always 360 okay so for example find the value of x in the diagram shown these are all exterior angles so if I add them all together I'm going to get 360 so X degrees plus 89 degrees plus 2x plus 85 you could put degrees or you don't really have to all of this will add up to 360 let's combine like terms X plus 2x gives me 3x 89 plus 85 174 and I'm going to say now I'm going to subtract 186 they actually kind of skipped a step in here you have 3x equals 186 divide both sides by 3 3 goes into 18 6 times goes into 6 twice so x equals 62 all right it's going to the last page the base of a lamp is in the shape of a regular 15 gone 15 gone has 15 sides find a the measure of each interior angle and B the measure of each exterior angle okay so two things we have to know right at the beginning first of all a 15 gone has 15 sides secondly a regular polygon hopefully you guys remember a regular polygon is equilateral and equiangular it's very important because you're going to see that that word regular a lot in these problems okay that means all the sides are equal and all the angles are equal okay so all 15 angles are congruent use the the polygon interior angles theorem to find the sum of the measures of the interior angles okay so we know there are equation and minus 2 times 180 in this case n is 15 15 minus 2 times 180 I'm going to use a calculator for this 15 minus 2 is 13 13 times 180 there we go 2314 okay then find the measure of one interior angle a regular 15 gone has 15 congruent interior angles divide 23 40 by 15 so two thousand three hundred and forty divided by 15 let's pull out that calculator again 156 okay so there are 15 angles each angle is 156 degrees so the measure of each interior angle in the 15 gun is 156 all right by the polygon exterior angles theorem the sum of the measures of the exterior angles 1 angle each vertex is 360 we don't have to plug it into any equation it's automatically 360 degrees so we're going to divide 360 by 15 360 divided by 15 see here 24 so the measure of each exterior angle in the 15 gone is 24 degrees all right I'll let you guys do the checkpoint and that's all