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8.5 & 8.6 Notes

Added Feb 16, 2015 •

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alright this is the last set of notes were going to do for this chapter eight five and eight six this is about trapezoids and kites the last time I learned all the different types of parallelograms the realm the rhombus the rectangle and the square now these are quadrilaterals they're four-sided but they are not parallelograms so they no one in the same category as the rhombus the square in the rectangle trapezoids or a quadrilateral that has exactly one pair of parallel sides and that is going to be this side and this side right here to be the this picture the top and the Bottom now the two sides that are not parallel or going to have a name in the two sides that are parallel or going to have a name the sides that are parallel are going to be your bases  okay and the sides that are not parallel or going to be licked this is going to be a leg in this is going to be like there's several parts of a trapezoid you have to know the legs the bases and then there's line that I have drawn through the center will not all trapezoids but you can throw it on all trapezoids and it is called the mid-segment we're going to learn some properties about it also the this thing you need to know is about angles here and the two angles that touch a base or called a base angle so you're enough to set singles but this single in the single together will be your base angles okay and then of course these two up here together will be a set of Base angles also  alright see you two sets of Base angles this ones that are along the legs don't have special night the mid-segment here's what the property is about the mid-segment the mid-segment of a trapezoid is parallel to each base and it's length is one half the sum of the links of the bases so right here what we have when this is your mid-segment if this face one in this is base to this is going to be half of Base one plus base to its links so it's basically average between the two bases so here's your property for any trapezoid there's really not a lot of properties but they do have a lot of parts by definition know that the bases  are parallel and by this theorem 8.17 about the mid-segment we know that the mid-segment will free ms is 1/2 base one plus base to that's what you need to know about a trapezoid the biggest thing is the top and bottom or parallel the mid say parallel and it's half of the bases added together there is a special type of trapezoid though that has more properties that we are learn about Nets and isosceles know it means the same thing as an isosceles triangle two sides are equal so what are the two sides will there the legs it's a trapezoid with congruent legs  so we have three theorems about this Durham 8.14 the trapezoid is isosceles then each pair of Base angles are congruent so are base angles are congruent okay let's go down here this is a base this is a base now base angles are along that so this angle is equal to that angle this angle is equal to that angle 8.15 if a trapezoid has  a pair of congruent base angles it is an isosceles trapezoid so that is just the converse  the base angles are congruent then it's isosceles what is Peru's it's isosceles 8.16 a trapezoid is isosceles if and only if that means it goes both ways it's diagonals are congruent so on your paper you have the diagonal drawn  AC and DB are your diagonal so let's go through the proper turn off saucily's trapezoid by definition the legs are congruent can you show this picture that would be a is congruent to BC those are my two legs by .14 and 8:15 remember those are the same thing just the converse of each other we know that they sing are congruent  K which in this picture would be angle a is congruent to angle B and angle D is congruent to angle C and then 8.6 the diagonals are congruent so diagonals congruent and in this case it would be AC is congruent to DB alright that's are trapezoids remember that  regular trapezoids all we know is that the bases are parallel to each other in the mid-segment is half of those added together and then isosceles we know that the legs are congruent so that we can mark that here in here we know that base angles are congruent and we know that the diagonals are congrats we have lots of stuff you're now one more thing I want to point out and it's not a property but you should know it because these are parallel to each other then what kind of pear is A & D together if these are parallel this would be a transversal so that would be consecutive interior angles soap these who's that are centered around the leg or going to equal 180 by parallel lines so that's how you prove that is because parallel lines not because of its Iran that they have here okay our third shape is a cut so we have trapezoids isosceles trapezoids then we have kites kites are not trapezoids it's it's own shape it's just a quadrilateral so we've learned about three types of quadrilaterals we've learned about parallelograms Kites and trapezoids so here's the cut kite is a quadrilateral that has two pair of consecutive congruent sides but opposite sides are not congruent so consecutive sides this side is congruent to that one and this one is congruent to this one that's the shape of a kite now it's the original kite shape that you're used to seeing but this is a specific shape I know not all kinds of shake this way but this is the kite shape alright then has to theorems to go along with it if a quadrilateral is a kite then it's diagonals are perpendicular so it's like a rhombus in that aspect that these diagonals are going to be perpendicular to each other all four going to be 90° so now we have four right triangles we can use our right triangle stuff to solve for things a point 19th if a quadrilateral is a kite then exactly one pair opposite angles are congruent only one pair of opposite angles are congruent which one it is but if you think about it there's only one set that could be congruent to each other and it's going to be this one and this one it's going to be the side between the one and the two Mark between the one to mark because that's going to one mark one Mart will not match up between two Mark to Mark see they have nothing in common so it's always going to be the angles between the one Mark and the to Mark the one market to Mark our properties One More Time by definition we have we have to pair of consecutive congruent sides which would be these two in these two then bite them 8.18 we have that the diagonals are perpendicular and by 8 point 19 we have one pair and only one pair of opposite angles congruent rapid right now let's go down here and let's come and look at this chart for just a second okay we learned about quadrilaterals quadrilaterals are for it's all four-sided figures no matter what the shape is and we have tons that don't fall into these three categories cuz you draw any kind of four sided figure specifically we learned about parallelograms first and we learned about two different types of parallelograms we learned about a rhombus and we learned about a rectangle  and then we also learned that if you put a rhombus and rectangle together you get a square then we learned about trap Dwight's right there and we only have one special type of trapezoid and it's isosceles paint and then wheel about kite so kite parallelograms and trapezoids are three different types of quadrilaterals rectangle rhombus and square are all parallelograms square is a rectangle Thomas and trapezoid we learned about isosceles trapezoids so that's how they all work you need to know the different properties for each and every one of these separately now class when you get to class on Tuesday we are going to put together a chart with all of their properties lined out so we will do that together in class some examples for all of these  play the first thing you're going to want to ask yourself is look at the shape if it doesn't tell you what shape it is in the direction you got to figure out what shape this is where it looks kind of like a trapezoid don't remember looks can be deceiving so don't always go by looks go by markings we know that that is parallel to that so one-sided well not too and we know that these two are so those would be the legs that are congruent and the bases parallel so we have an isosceles trapezoid so what do we know about isosceles trapezoids well we know that the base angles are congruent so yes this one is 110 what is h have to be well it's going to have to be 1:10 alright then we still if on fmg so those are not congruent to these however if these are parallel this is a transversal and which would be a pair of consecutive interior angles and we do know that they have to add up to 180 180 - 110 would leave you 70 degrees each one of these is 70 degrees cast OnePlus base to okay well that's a base that's a base atom together so you would do one half of 21 plus 19 so 21 + 17 is 38 half of that would be 19 so this would have to be nice look at the next one jklm is a kite so we know it's a kite think about the properties of a kite it says just find angle alright well remember the angles that are between the one Mark and the two Mark have to be equal so angle K is just going to be 88 degrees let me take that a step further if you had to find J could you do it now think about it just for a second what would you do you know that l&j or not equal only can you no more but it is a quadrilateral how many degrees are in a quadrilateral 360 degrees so you can add the three that you know the 8 the 88 in the 120 and subtract from 360 and you're going to find out that this would have to be 64 if you had to find it which will be in a lighter alright let's look at the next one's okay number 12 it just says find the value of x again it does not tell you what kind of shape this yes so let's look at it for a second parallel parallel not parallel trapezoid now is that isosceles well look at it very carefully that's one more than one mark in the mid-segment does cut the sides and half I don't know if we really went to detail about that but talk about that right now A mid-segment is from midpoint to midpoint I don't know if we did the formal definition midpoint two midpoints what does cut this sudden half in this sudden half but these are Mark the same as these so therefore those legs are not congruent but we do know it's a mid-segment and we know that this is equal to have these two added together so think about that and write that out as a formula I do know that 1/2 of 10 plus 44 s equal the mid-segment which is 2 x -1 now just do some math 1/2 of 10 + 44 is going to give you 27 equals 2x minus 1 add 1 to the sides you get to x equals 28 cuz those cancel so divided by 2x equals 14 so we do know that X is 14 here 15 is definitely a kite these two equal these two are equal so what do we know about this single and the single they are congruent so your who's going to set those equal to x - 111 not - equals 111 sorry / 2 x is equal to 85.5 and that's your ex you could plug that again you should get 111 right here correct and then you could add 111 111 and together sheetrock from 360 if you had to find that bottom angle  okay this part is going to come from that 8.6 section where we turn all together we try to identify things and it says give the most specific name for the quadrilateral so they give you that it's four sided but you got to tell me why it is a specific thing and get the most specific one that you can so let's look at this one okay we know that the top and bottom are parallel but we also know that one pair of Base angles are equal now we had a theremin told us if base angles are equal then we had a specific shape and what is an isosceles trapezoid so this one would be an isosceles  trapezoid k  alright number wait now we have 3 right angles and all sides are equal but think about it if this is a quadrilateral 90 90 90 this one has to be 90 to make 360 so I got 4 right angles and I've got four congruent sides but does that make that has to be typically a square alright look at number nine number nine gives us that we have those congruent those congruent and then angles across from each other congruent so if you look at your theorems and definition that has to be a fight and number 10 now it looks like a rectangle but do we know is it a rectangle all they tell us is that parallel and these are parallel opposite sides are parallel don't know angles we don't know side measures so the only specific thing that we can call this one is a parallelogram cuz parallelogram has opposite sides congruent and parallel Pacific things that you can say already on your paper remember we'll do a chart when you get to class on Tuesday and go over this questions just

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8.5 & 8.6 Notes

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