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Section 6.6 Notes and Video

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that's what I want to look at our last section which is going to cover trapezoids and kites so the last two quadrilaterals we're going to talk about if you remember our figured that I drew up on the board last week we had the umbrella of quadrilaterals and parallelograms which included rectangles and rhombi and then that's because both of these were things that have characteristics of a quadrilateral they are closed figures four sided polygons but they are do not have the same characteristics as each other or is parallelogram so they're kind of on that horizontal same row as a parallelogram the trapezoids for a characteristics for trapezoids first two trapezoids first so the trapezoid characteristics are this there is exactly one pair of parallel sides those parallel sides are called bases and in the nonparallel sides are called legs so for instance if I get a trapezoid here and it looks like this this top base and this bottom base would be all the bases they would be parallel but not congruent and then these two non parallel sides would be called the legs if the legs of a trapezoid are congruent so if these two saw parallel if they are congruent then that is also called an isosceles trapezoid so it will be a special type of trapezoid to be characterized as a trapezoid it does not have to be isosceles but if it is isosceles extra characteristic one is at the legs are congruent secondly that that's oscillate is that the base angles of the angles that are on the bases different than a parallelogram opposite angles it would be these angles at the the bottom or detangle to at the top would be congruent finally just like a rectangle and I saucily's triangle the diagonals are congruent another thing that we have with trapezoid is we have something called the mid-segment and the mid-segment of a trapezoid is parallel to each of the bases so Francis if I have the trapezoid trapezoid ABCD for instant video that would be the mid side so the mid-segment the equation of that is that the mid-segment is half of the sum of the bases so in this particular example would equal one half of the sum of what's a date so of Abby was Qin and CD was 20 I would add 10 + 20 + 30 + so half of that would be the 15th that we have would be 15 so don't like seeing some examples here the number one is an inside of a basket shown is an isosceles trapezoid if the measure of angle jml so jml is this angle right here if that measure is 130 degrees it said that kn is 6.7 bait and M in is 3.6 feet it wants us to answer to question first it was is the measure of mjk so if we look at in JK if we talked about remember our are characteristics of trapezoid sit at base angles working group what this is a trapezoid the bases would be in L & J K Atwood baby parallel lines so Emma will be congruent meaning of both 130° J&K would be congrats so therefore it is a Mandel are congruent and the relationship between M that we know is 130 and Jay which is what they're asking us to when is a bear supplementary so we would add a hundred 32 degrees plus the measure angle mjk 2 equal 180 degrees for the measure of angle mjk what equal 50 degrees and you can obviously use the symbols for angle the little mm rope from measuring the degrees symbol I just can't write that on this Maxwell concert on Jal Jal is a bag of a wolf this is an isosceles trapezoid diagonals are congruent so therefore we know that the measure of Katie is going to equal to the measure of JL we know that KTM is 3.6 b + 6.70 + so that equals jail so therefore Jal is going to equal 10.3 8 what we have and I thought we are they figure why we have a quadrilateral ABCD given these vertices 5 one negative 3 negative 1 2 3 Sorry Brad on your sheet there and ask to show that if the ABCD is a trapezoid and determine whether it is an isosceles trapezoid so slow first to determine if the side of your had one pair of only characteristic of a trapezoid has one pair of parallel sides obviously looks like CD and a beer those pairs what we want to do the slope of all in order to figure out if it is if the office God are parallel and which ones are if they are so CD slope if we plug into the slope formula we would have 4 - -2 better by 2/4 minus 3 divided by 2 minus negative 2 Wychwood simplified to give us one for then we would do a B2B would be -1 -1 divided by -3 minus 5 which would give us a slope of again 1/4 then I would also check my slopes of c d c b and a d just to confirm the trapezoid is not a parallelogram or any other type of thing so see me would have a slope of 3 - -1 divided by -2 - -3 which is the slope of 4 and Natalie 80 would have a slope of 4 - 1/2 - 5 which is a slope of negative so what we see here is that CD is parallel to a b lady is parallel to a b that one pair of opposites side is parallel therefore it is for the second part of the question asked us to confirm if it's an isosceles trapezoid so I pray that it's a trapezoid one pair of opposite sides parallel and I saw a flea trap and we can do that by using the characteristic that diagonals are congruent in an isosceles trapezoid or we could check the legs be the basis of their parallel you could either do diagonals being congruent or which would be AC and BD or we could do the leg then can they see it so I was going to do the legs one of them will prove that it is or is not an isosceles trapezoid so we did the distance formula of a thing we're going to say that it comes out to be a radical 17 and of course you can go and do this later on just for the purposes of right now I'm just giving you the answer but you do know how to do the distance formula and you will need to do that turns out to be 3 radical 2 so a dude's distance turned out to be 3 radical 2 because they are not congruent because BC is not congruent to ad the Fig legs are not congruent therefore it is not and isosceles trapezoid so we can pray that it's a trapezoid because we stayed that 80 and CD are parallel and one pair of opposite sides parallel proves it's just a trapezoid but it's not an isosceles trapezoid look at the legs are not congruent now had you done the did you also try the diagonals you would have seen the diagonals are not congruent therefore it's not an isosceles trapezoid either one of those would work of course you stay here exactly what we're saying okay for number 3 it's asking us to find alright for number 3 that I can use to find the length of X it says that A Min is the midst the trapezoid remember we talked about earlier that the mid-segment in this case in in equals one half the sum of the bases so in this case would be KF play JG in case we know a man is 30 and that's going to be one half of KF is this 20 + x which is JG and I'll show you both we can either go ahead move the two to The Other Side by multiplying by the reciprocal of 1/2 which is 2 and we would get 60 equals 20 plus tax and then we subtract 20 we would get X is 40 Eunice and if we check 20 and 40 added together / 2 does give you 30 so that's always a good thing to check now the other way we might could have done it after we set it up 30 pulls 1/2 of 20 plus tax we could just give me the one half which would give us 30 equals 10 + 1/2 x charge for 10 and get 20 so 20 equals 1/2 x and then x 2 and still get 40 equal but I want you to try the next one so Athens 5 CD is 35 and I want you to find B okay so to get bet we take 1/2 of AF which is 5 + CD which is 35 so that would be Bes equaling one half of 40 so be what equal 20 status pretty straightforward also keep in mind that abef is also trapezoids and you can even have a good thing that there I could even say find the mid-segment let's just say that I draw went in and they find the mid-segment of XY well now that you know that bet is 20 you can add 5 + 20 divided by 2 and get that X was 12.5 or I can have you find the midpoint of this trapezoid because this keep in mind you can have trapezoids within trapezoids when we have a mid-segment thanks for going to talk about kites are characteristics of kite or this diagonals and kites are perpendicular not congruent they are purple radicular so if I see if I can figure out here's one if I have a kite for instance and I will just take this figure here the diagonals and let me see if I can put some wine for the middle of this so the diagonals here would be perpendicular to each other so what would happen so what would happen here is at the intersection of these diagonals in just two kind of so I can refer to and it make it easy I'll put that this is kite a d c d and I'll put the so Ed intersection to eat V AC and BD are perpendicular at the intersection of e that means I'm going to have four right angles right there at 7:04 right triangles know these right triangles are not go be congruent because these triangles up here maybe shorter or greater than these down here so the other characteristics is that exactly one pair of opposite angles is congruent Francis and nutcase let me just kind of mood this up a second so if you look at the picture now it looks a little bit better actually so we say the things are for right triangles so are Pythagorean theorem can happen here that sort of thing cuz they're bright on go through the intersection but days to talk to the next to bottom triangles are congruent other so it's not that all four congruent like we would say maybe in like a rhombus or something exactly one pair of opposite angles that have opposite angles A and B. Always the case depends on which way the flipped but in this case and it finally exactly two pairs of consecutive congruent sides so I'm like a parallelogram with opposite sides are congruent we do have two pairs of congruent sides so it's different than a trapezoid I'm here with two pairs of congruent sides but those who resides would be a b and a d and in DC and BC so we would have that these are equal to each other these are equal to each other we would have that this is going to actually be equal to each other because their bisecting opposite angles bisect opposite angles to not only is DND equal but it's being bisected so both of them are congruent outside the whole Tangled A and B are congruent and bisect it but the whole angled and beer can grow and hiring will not not necessarily the by section but these triangles are congruent by SAS or you could even think they're going to grow up by I guess our FSA office or any of those different ways that we talked about look at some examples I'm going to do number 5 8 and then I'm going to give you the answer to me and I want you to try it later on your own so that you have it so in this particular problem what which says we said that if that WXYZ is a kite and I want you to find the measure of angle XYZ so we know it's two sides or angles are congruent based on the location of the congrats so since w x and w we are congruent that mean that the angle between that W is the non congruent angle to why so it has to do the two congruent angles have to be the angles that are between the non congruent sides so that means that if 12010 is also 121 down the beginning of this chapter we talked about the interior angle that we talked about every quadrilateral has two triangles within it there for 180 x to the G36 so in this case 73 + 121 + 121 + angle the measure of angle why is going to equal 360 degrees play subtract 7321 + 121 / 360 we get the measure of angle y or XYZ to be 45 degrees now this second example we see something very similar we stayed at D&B would be so we would add 120 85 + 85 + angle C to equal 360 and we would get that the measure of Anglesey is going to equal 70 degrees and I would want you to try that one later to make sure you do come up with that the final problem we're going to have with a kite is one that we talked about the size remember we know about the side is that a D&C it would be congruent a B and B stay would be congruent and also we know that this line AC is being bisected by DJ and we have some right Tangles hair at the intersection remember that was one of our big one look at triangle ATB in the intersection that doesn't really have a letter will just call it E2 so that way we can talk about it we see that right angle and we see that 80 is 4 + B is 3 will we can do the Pythagorean theorem here right so we can to find a bee print since we can save for square plus 3 squared equals 80 Square and thought we would get 16 + 9 equals 80 Square so that would be 25 table I be square we would take the square root 25 and 80 and that would give us that ad would equal 5 units Pythagorean triple we would know that I maybe was 5 years just because we need to the 3 and the 4 for the Lex now one thing I also keep in mind from the e-2c is also for you buy it from DTE is not 3 what's the non congruent one that is the diagonal that is not being bisected only one diagonals being bisected in a kite and that is the diagonal that goes through the angles that are congruent to each other and in the whole angles congruent angles no I don't want you to try one more try this last one right here they want you to find as we talked about earlier remember your two congruent angles in this case are going to beat him and pee because there between the non congruent sides so therefore RP is also going to be a unit and if we see that are in is 6 and RPM gauge to find NP we can do the Pythagorean theorem and get that mp10 if you remember the last talk about that 3 4 5 triangle 3 x 2 + 6 for X Series 800 5 * 2 is 10 so it's kind of like a play off of that trying so next what you're going to do is going to be giving your homework sheet and Groupon want you to work on your homework I am going to post this video to my website in case you need to watch it again and I will see you guys tomorrow remember your test is on Thursday so you need to make sure that you're working on not only your exam review for your test next week that I'll give you tomorrow but you're working on your exam or before this test on Thursday I'll see you guys tomorrow

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