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8.1 & 8.3 Notes

Added Feb 10, 2015 •

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alright guys chapter 8 notice that it is not triangles it is quadrilaterals in this one but we will make triangles out and somehow quick review polygon remember that polygons are closed figures made by three or more sides and we learn that earlier this year remember also the difference between concave and convex and remember the technical definition concave if you can extend one of the lines and it goes through the figure it concave convex if you extend all the sides none of them will go through the interior and that's what makes it convex convex is out concave is caved in somewhere now do you remember the names for all of the sides 3 through in let's go over those for a quick review okay remember if 3 sided figure is a try a four-sided figure is a quadrilateral 5 sides is a pentagon 6 is a hexagon 7 is a heptagon 8 is an octagon 9 is a nonagon 10 decagon 12 dodecagon and any other thing you'll put the number Dash Ganso 14-gon 12 well we have 12:20 or gone and so forth couple more things that you need to remember remember the word regular and remember regular is a big word here when you see a regular pentagon Turn The Irregular heptagon a regular octagon a regular decagon whatever it means that all the sides and angles are congruent change and you need to know all the parts of a polygon the first thing you need to know is the vertex the vertex is all corner so on this one there's going to be one two three four five six and those are all a Vertex or vertice the second thing you need to know is what the sides are and of course the sides are from vertice to vertice so this one has 1 2 3 4 5 6 sides which makes it a hexagon are the diagonals this is one diagonal the blue line and it's from any vertice to another vertice that's not adjacent to it so this would be a diagonal this would be a die tell now you can draw a diagonal from any vertice to another one when I say that it can't be adjacent if I come from this one I can't go to that one cuz that just makes a side knot diagonal so these are also known as your diagonals  all right now the next thing you need to know is Define how many degrees are in each kind of polygon know if you'll know this this is only for convex polygons okay it's called some of the in angles of a polygon and it's a theorem 8.1 and what it says is that the sum of the interior angles of a polygon with n sides is in minus 2 x 180 degrees so you take the number of sides and you subtract two numbers add subtract two and then multiply that amount by Indiana roll tell you how many degrees are in a polygon so we already know that a triangle has 180 degrees and it will think about it just got three sides three close to is 11 * 180 is 190 think about any four sided figure 4-2 is - 2 x 180 is 300 50° and that's where we get that from  now 8.1 was for the sum of the interior angles all the inside angles eight point who is the sum of the exterior angles so you'll see exterior angle theorem the sum of the measures of the exterior angles of a convex polygon one at each text is always 360 Degrees no matter how many sides there are the outside angles add up to 360 degrees and when I say wanted each vertex what you would do and it's on your paper you extend each side and I know this is not Flyknit up but it would be this single plus this angle + this single + this single + this single though add up to 360 Degrees I know that looks really bad alright so next thing is a couple of examples to go with that in this is really kind of is he here guy so don't make this hard okay so find the sum of the interior angles and the exterior angles for each polygon no remember it's in -2 180 is the interior and -2 * 180 any xteriors always 360 so we have an that would be 8 - 2 x 180 degrees so that 6 x 180 degrees which is 1080 degrees total that's the interior now the exterior is always 360 degrees this is in rear that's exterior okay 17 argon 17 side so you would do 17 - 2 x 180 degrees which is 18 x 180 degrees which is 2700 degrees that's the interior okay the sum of the exterior is 360° doesn't matter how many sides the outside adds up to 360 because there's you add more vertice the outside angles gets smaller and smaller and smaller and that's how it always add up to 360 alright now look at example 2  okay it says fun value of x and y in this polygon so if you'll notice it has six sides so that's a hexagon so you need to find the total first so you'll do 6 what's 2 * 180 degrees which is 720 degrees total so all of these inside angles have to add up to 7 20 so if I take 720 and subtract all the ones that we know we should get the missing angle so we do 720 degrees at all those up 656 degrees subtract that and you get 64° that's my ex so this is 64 now think about it how would I why well here's a linear pair so they have to add up to 180 so do 180 - 64 degrees and you 116 degrees for your why so this would be 116  okay example 3 the sum of the measures of the interior angles of a convex polygon is 2340 so that's the sum classify the polygons by the number of sides so how many sides would it take give me 2340 degrees inside well your formula is in -2 x 180 degrees so that equals the Sun so that would equal 2340 now just solve algebraically for your in which is your number of sides now the first thing I would do is I would defy by 180 degrees now you can distribute it and then do it that way but those are both multiples and they cross out sew-in -2 equals  13 2340 / 180 s13 now Pad 2 to both sides and you 10 equals 15 so 15 sides will give you 2340 and you can test it 15 - 2 x 180 you get 2340 find the measure of the interior angle and an exterior angle so and we're looking for one of a regular heptagon have 7 regular means all the sides and angles are equal so if we know the total we can just divided by the number of angles and that will tell us how much each one is so thank seven sides would be 7 - 2 x 180 degrees 7 - 2 x 180 is 900 Degrees so there's $900 is going to have to gone so 900° what's a 0 we / 7 7 sides 7 angles that would end up being a 28.6 degrees for the each interior angle Kent's now each x 2 Google all the exterior always adds up to 360 so I've divided by 7 cuz there's going to be 7 angles and that gives me 51 4 degrees  not your exterior skating think about it these two just like that problem up above right here sorry the exterior in the insurer have to add up to 180 if it's on the same corner so right here since they're all equal because it's regular these have to equal 180 degrees which one 28.6 and 51.4 **** set you could have also found the exterior by doing what ladybugs the interior  okay into your son's extra sums that was 8.1 in this Pages 8.2 and 3 this left column is 8.2 finish right column is 8.3 and they go hand-in-hand and that's why I put them together in the same lesson now we're going to learn all the properties of a parallelogram not a parallelogram is a your lateral so we're going to talk about quadrilaterals in this chapter so if you look at it if you have a parallelogram there are four properties that you know okay so if you have a property a parallelogram 8.3 tells us that the opposite sides are congruent so opposite sides are congruent if you have a parallelogram going to go down to this parallelogram what that would look like is that all your opposite sides are congruent so this side is congruent to this side this side is going to be congruent to that side okay 8.4 tales as if you have a parallelogram then the opposite angles are congruent so what that's going to look like down here is this angle is going to be congruent to this angle and this angle will be congruent to that now this is all parallelograms will have these properties okay now 8.5 says if you have a parallelogram consecutive angles equal 180 degrees so if you have a parallelogram the consecutive angles equal 180 consecutive means one right after another soap these two together equal 180° or these two together equal 180° or these two together equal 190 degrees or these two together equal 190 degrees so any set right after the other equal 180° and opposites are going to be equal now the last one have is theorem 8.6 and that tells us if we have a parallelogram then the diagonals bisect each other no I don't know how draw the diagonals on here let me try  play there's one and here's the other so those are the diagonals now they're going to bisect each other what that means is this one and this one are congruent and this one and this one are congruent because diagonals bisect the other so that's everything you need to know about a parallelogram okay opposite sides are equal opposite angles are equal consecutive angles equal 180 and the diagonals bisect each other all right now that's if you parallelogram you have all of those things now on the other side  on this side if you know then you have a parallelogram so this is all of these are the converse and pretty much all four of them match up over here except for this last one so tell you prove you have a parallelogram you have to have one of these for situations number 18.7 if you know that both pair of opposite sides are congruent then you have a parallelogram  so if nothing is marked on the quadrilateral except that the opposite sides are equal then you know that's enough information to tell you that you have a parallelogram and all the rest will be true also okay number to there an 8 point it says if you know that both pair of opposite angles are congruent then you have a parallelogram  so on this picture down here if the only thing you have marked with this angle is congruent to that angle in this angle is congruent to that angle that is enough information to say that has to be a parallelogram and you know everything else now do you see how these too much up and how these to match up alright serum 8.9 says that if one pair of opposite sides are congruent and parallel that is enough to tell you you have a parallelogram play this is the one that does not match up across right here cuz he this one says consecutive angles are 180 this one says and it kind of does but it doesn't one pair turn see both of this is both pair both pair one pair of opposite sides are congruent and parallel so it can be these two you know our cool and parallel or these two are equal and parallel and I think somehow we missed somewhere up here I'm sorry that's my fault that if it's a parallelogram by definition the opposite sides have to be parallel so this is parallel to this side and this side is parallel to this side opposite sides have to be parallel last one is that theorem turn in what it says is if the diagonals bisect each other then you know it's a parallelogram so see it matches up with 8.6 make here so let's write that out okay so that's how you can prove that it's a parallelogram or if you know it's a parallelogram all of it attribute so remember opposite sides are parallel and congruent opposite angles are congruent consecutive angles equal and diagonals bisect each other you need to know that last page is some examples using those properties and you might want to pause it and see if you can figure out put on before that you actually play it and see if you got it right okay number one find the value of each variable in the parallelogram now if I know it's a parallelogram the number one I know that the opposite sides are congruent so these two are equal to each other in those two are equal to so if I said these two equal I can find lie  so why subtract 2 from both sides and why is 14  then two I can set equal to each other because opposite sides are equal so 2X equals 10 so x equals 5 and that your answer number for please two angles have in common where their opposites so they're equal so 4 m equals 52 / size by 4 and M equals so there's your answer alright then let's move down okay now look at this problem before you start it is a parallelogram by that symbol right there it's telling catsuit fun in the indicated measure so remember diagonals bisect each other so those two are equal and those two or equal so 3M is going to 415 that diagonal and 12in or 12-10 is going to equal to in because those are the two parts of that diagonally I'm 5 and 6 5 says Fine PR PR is this entire distance well if you're going to be smart about it and think if I want the distance from Pete can I know that half is 15 this has to be 15 sopr is 30 now you can set 3 mm equal to 15 and solve but they just want PR and they give it to you now stst is only half of that it's from here to here now we got a soft for in to find that so if we set this equal that we can find in so-so 12 minus n equals 2 in at the end over here so those will cancel and I have 12 equals 3 in / sides by 3 and in is going to equal for now I've got to find s t so go back over here to St can you put A4 right there 12-4 is 82 St is eight angles we need to find SRQ so SRQ that's this angle will this angle is 2x the whole thing and the only other angle they give us is this song the whole thing now these are consecutive angles there one right after the other so we know they have to add up to 180 degrees so if I do 5 6 + 1 + 2x right here it's got to equal 180 degrees now just do the algebra with that so if you distribute the five get 2x + 5 x + 5 equals 180 which is 7x plus 5 equals 180 degrees subtract the five over here 7x equals 175 divided by 7 and X is equal to 25 now we have to find SRQ so SRQ is 2x so too times 25 is 50 degrees so 50 degrees is SRQ and that's that problem right there now look at pqr if I need to find pqr in I already know SRQ pqr this angle these two have to add up to under 90° there consecutive so all I have to do with an already know that's 50 I can subtract from 180 to find that one so you would just do a hundred eighty degrees minus 50 degrees equals 130 degrees and you are done with pqr okay the last set of problems which theorem can be to show that the quadrilateral is a parallelogram now if you look at the page before these are all the ones on the right through mate 7 88898 10th number one what are you giving will look that is equal to that and that's equal to that the diagonals bisect each other so if you look over there theorem 8.10 says diagonals bisect each other then it's parallelogram so this one is going to be serum 8.10 number to we have that these side that opposite where is equal and then we have that these two are equal so opposite sides are equal that's 8.7  both pair of Ops add your equal number 3 look at it we have that these two angles are opposite and equal and we have that these two angles are opposite and equal so if you look at 8.8 both pair of awe angles are equal so theorem 8.8 tells me this has to be a parallelogram and last but not least 14 these are parallel end their equal so one pair is parallel and equal opposite sides that esterom 8.9 that says  and that's all you have to do alright your assignment okay page 510 3 you're 27 multiples of three that's not very many page 518 4 through 16 multiples of 4 every fourth problem 23328 that's five more in than 40 and then the last one is 526 number 4 through 16 and that's just name in the theorems I think 15 through 17 25 and 34 I know it looks like a lot but it really is not okay we're going to try and make this due by the end of the day Wednesday if possible we may have to play it by ear a little

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8.1 & 8.3 Notes

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