Thank you, Mr.. True in this video. We are going to take a look at
Trapezoids now [follow] my geometry videos been talking a lot about special types of parallelograms a
Trapezoid is not a parallelogram
[Robinsons] rectangles and squares those are types of parallelograms where you have two pair of opposite sides?
Which are parallel and trapezoid you [only] have one pair of sides which are parallel in this video?
We're going to do one pretty simple example here. We're going to do another example
That's got some Algebra involved in it with the mid-segment theorem, and then we're going to prove
the fact of diagonals of an ISosceles triangle
[to] me Trapezoid are congruent at the end of this video
okay, so
Type is always again [our] quadrilaterals. That's a straight sided figure with four sides
[and] again only has one pair [of] opposite signs which are parallel
[here] we have a general trapezoid
[the] sides the two sides which are going to be parallel [to] each other are going to be the bases
It's not necessarily the top and bottom if you will I can turn this orientation around and have
My parallel sides maybe going this way, and you can just simply turn your book or redraw it on your own piece of paper
So [that] the bases are horizontal. It's kind of standard, but certainly not required
the two parallel sides are
The bases the two sides which are not parallel are always called the legs regardless of whether again
They're kind of going up in an angle or you know however
I have this parallelogram situated the parallel sides of the bases the two other sides
opposite sides are the legs
Every angle inside of a trapezoid is going to be considered a base angle because they're all going to be against you know one of
The bases whether [that's] this base or the space?
Our examples are mainly going to be focused around the ISosceles Trapezoid. Which is like a triangle
You know the real definition of an iSosceles triangle is a triangle with at least two sides which are pair up congruent
But if we have all three sides, we generally call it an equilateral or equal angular
triangle
Like an ISosceles triangle has two congruent sides and Isosceles
Trapezoid
Has the legs which are congruent I could probably draw this so that one of the bases
is the same length as the two legs, but that's
You [know] that's not what we're [looking] at
we'll look at Anna sausage triangle and trying to Define it the two legs are going to be congruent, so
if you think of this
This Isosceles trapezoid [as] an ISosceles triangle that we just chopped the top off remember that in an ISosceles triangle
The base Angles are going to be congruent. Well that same thing holds [true] here
If a quadrilateral is an ISosceles trapezoid then each pair of the base Angles are going [to] be congruent
So these two legs are congruent. So these base angles that are opposite those
Legs which are congruent are themselves fronts, so these two are going?
Opposite the legs which are congruent and these two angles are going to be congruent
the consecutive Angles
along each of the legs can Vary well most the time we'll have different measurements, but
You know here's the bottom base those were on top base those are growing also the diagonals of an iSosceles?
Trapezoid are congruent themselves
so if I have an ISosceles trapezoid
And I draw a diagonal going to each of the opposite [vertices] on those lengths are going to be equal
We'll be proving this particular Theorem or property at the end of this video
an example
So I have a quadrilateral. [I] know it's a trapezoid because again
I have one pair of opposite sides which are parallel and
I want to find the measurements of all of the angles well
I have these little tick marks on each of the leg which means that this is an ISosceles Trapezoid on't have to necessarily write that
Down [you're] suppose to be able to recognize that and the base angles of an ISosceles Trapezoid are congruent
[so] this is 65 degrees guess what the measure of angle three is and again
We're [talking] about the angles that are opposite those legs which are congruent
So those angles those base Angles are going to be equal the measure of Angle 3 is equal to um
65 degrees
Okay
Well, what's the measure of Angle [1] if I can find the measure of Angle 1 I will also know the measure of Angle 2
well
You know can't forget all those properties we've been learning in Geometry every one of these Chapters Builds upon themselves
And when you have two parallel [lines] that are being cut by a transversal
[those] uh Miss [down] a little bit, but those same side interior Angles are
supplementary
so the measure of Angle 1 plus 65 degrees
Must be equal to 180 degrees supplementary has two Angles which add up to 180
So we have a little equation to solve for it's very simple. We're just going to subtract both sides by 65
and
The measure of Angle 1 is equal to well 180 minus 60 is
120 and 120 minus 5 is 115 degrees
So the measure of angle [one] is 115 degrees the measure of Angle [3] is 65 degrees
and we can make sure that's right by adding these two and make sure they do add up to 180 and
Again Angle 2 is a base angle a congruent base angle to the measure of angle [one] so it's also going to be 115 degrees
Trapezoid mid-segment theorem we talked about midsegments and triangles they had special properties [that] when you had
Any triangle didn't have to be ISosceles, but if you had a triangle
And you drew a mid segment that that mid segment was parallel to the third side
Of that triangle, and it was equal to half the length. We still have a good
relationship
to our trapezoids
in a trapezoid if you have a mid segment a segment connecting the
Midpoints of the opposite or the legs [I] have this kind of looking like an ISosceles trapezoid
I just I don't know why I drew it that way. It doesn't have to be
That's why I have the single two marks over here that this is a mid segment or midpoint because these two segments are equal
Midpoint because these two segments are equal these do not have to be equal to each other on the opposite sides of the trapezoid
The mid segment is parallel to the [bases], okay?
so I have a trapezoid [one] two pair of parallel or tuning one pair of
Parallel sides and that mid segment going through the middle of the trapezoid is going to be also parallel, so we have three parallel
segments
The length of the mid segment that blue line is equal to half of the sum of the lengths of the bases
We're in a trap in a triangle
We only have one side that was parallel, so the mid segment was half of that length
[we] still have this half thing going on, but we have two bases
So we're going to add up those two bases the length of a B plus the length of DC
take that sum and divide it by 2 or multiply it by 1/2 same thing and you are going to get the length of
the mid segment [oops] forgot my letters my mid segment of
Ef let's get to an example that's dealing with this theorem soon as I find my remote
So here we have a trapezoid
[why] is it a trapezoid because I have one pair [of] parallel sides?
We have a mid segment going through the middle of it
Why because [I] have?
well tick marks on each of the links shown they've been cut in half and thus ef is the mid segment of
Trapezoid
Abcd which means that the length of this
Mid segment is equal to half of the sum of my two bases, and I've rotated this a little bit
So you don't think the bases are always horizontal?
Okay, so we have to solve this for
What solve for x and we'll find the three lengths, so we have 4x plus?
5 is Equal to 1/2 the [sum] of the other two
of the basis, so we have
4x minus 7 plus 6x
minus
5 I picked one with some you know actually [good] right here
So we need to solve this for x we need to of course work our way out of the [parentheses]
So we're going to combine all these like terms here. We get 4x
Plus 5 equals 1/2
We have parentheses in [here] always do it with substitution
Just to make sure I don't have any issues with signs like if this were a negative if I were
Subtracting the [bases] for some reason I would you know want to remind myself to actually distribute that negative?
Through the parentheses if there was one or exponents, whatever but um there's no exponent. There's no leading coefficient
That's other anything other than [1]
There's no division going on so these parentheses are just my habit of how I?
Substitute. They're not really going to do anything because nothing can be added in here or in here, so we have 4 [plus]
6x which is equal to 10x
We have negative 7 plus negative 5 so we want to add those and keep the signs. It's negative 12 and
Now that we've got the inside of the parenthesis cleaned up
We're going to take this 1/2 and distribute through the parental distribute through the parentheses
And we get 4x plus 5 is equal to half of 10 [which] is
5x and half of 12 which is negative [simple] negative 12 which is negative 6 and
We got you know just a simple linear
equation with only one variable so we're just going to solve for x we're going to
Let's see I'm going to subtract [both] sides by 4x
and
Get 5 is equal to 5 minus 4 which is 1 x minus 6 we're going to move that
X on the right hand [side] now, so I'm going to move everything over to the left
That's this left which means I'm going to add 6 to both sides and get
X is equal to 11 which is the same thing. I got one in [my] notes now
It's even if the direction said just solve for x or
If the directions didn't say solve for x if I wrote an e I'd be done
But if I need the actual lengths of all these segments, then I'm not finished
I want to take this value of 11 and plug it back in here and also
It's not a bad idea even if a question says solve for x to take it plug it into these
Expressions and actually find out if the mid segment is half of the son of the two bases [you] know just to check your work
Even if I'd only needed to find x so we [have] let's see here. We have that a b is
Equal to 4 times x which is 11
minus 7 well that's going to be equal [to] 4 times 11 is 44 and
44 minus 7 is going to be equal to let's see here. How about 37?
Okay, the other base is let's just go in order e f is equal [to]
I'll put a starter and y myself at the [mid-segment] is equal to 4 times x which is 11
Plus 5. So that's 44 again
Plus 5 is 49 and we have DC
Which is 6 times x?
minus 5
Yeah
That's 66
minus 5 is equal to 61 and
Let's make [sure] those are right. Let's see here. We have 49 should be equal to 1/2 of
37 Plus 61 that's 49 is equal to one half of
sixty one ninety one [ninety] eight and
49 Let's See here half of 98 Well 98 Divided by 2
98 Divided by 2 Well 4 times
2 is Equal to 8 so we have that?
8 a ranger of 1 so bring down [that] they got 118
18 divided by 2 is equal to 9 and that means that the [mid-segment] was half the sum of the two bases added together
So now not only do I know my no no my three segments
But I know my answer is right because I have checked it. We're going to prove that the diagonals of an ISosceles trapezoid
Are congruent the two column proof?
triangle or excuse me Trapezoid Abcd is a
Saucily so we're going to prove that the diagonals are congruent that Ac is [congruent] [to]
Bd
Statement reason two Column proof trapezoids ISosceles, I just said that so it's given
Okay, now you can see that as we've drawn our
Diagonals through this trapezoid. I'm actually going to prove similar to this in a previous video proving congruence with triangles that overlapping
Couldn't use properties of trapezoids in those because I hadn't used them yet
But I can now we have triangle acd and triangle
d be a make sure you [make] sure remember when you name your congruent polygons that you keep the corresponding angles in the
Proper order of those names if I can prove that those two triangles are congruent
Then well congruent parts of your on triangles or grunt thus [we'll] be able to say that Ac is
Congruent to bd. But for now. I don't know that [those] two triangles are congruent, so let's just start with you know
Trying to build up that information that we need
proving two triangles and [garant] side side side side angle side angle side Angle
or angle angle [side] unless they are right triangles, and then you can sneak away with that hypotenuse leg thing, but
Yeah, we need to build that information on our prove so the first thing I can see here. Is that b a is congruent to
Cd
Because well they're the legs of an ISosceles trapezoid and thus they are congruent based on that definition
There's the lights they're congruent because like I said
we know
Legs of my sauces trapezoids are congruent
Okay, well again
We're trying to build up information about these two triangles enough to be able to say they're congruent and these two triangles
[I]
brought up the showing congruence or proven
congruence and overlapping triangles
Because well they overlap and they share a [side] and every time
Two triangles share a side somewhere in your proof you're going to be using something called
HmM starts with an [arm]
Yes, that was the reflexive property of congruence ad is congruent to ad that side that is shared in both of the triangles?
That's why we have to write the same thing twice saying that it's equal to itself
It's [caused] as part of two separate triangles that we're trying to show are congruent
Well, let's mark that in our diagram here
So I'm really close to using let's see here. I think I'm just close to using
Side-Angle-side uh the angle be a d
Which is making up you know of course part of this triangle here bd a and Angle [C]
D a is part of this triangle here if I could somehow say that these two angles are congruent then I could use
think
Side-angle-side to prove my triangles are congruent. How can I prove that these two base angles of an ISosceles?
Trapezoid are congruent. [I] think we just learned that in a previous theorem
And that's going to use to verify or state that those two angles actually are congruent
Well my base angles are congruent. I kind of gave away my next line already
[these] two triangles are going to be congruent due to the side angle side theorem
[just] make sure you again you get those some letters in the correct order if you wanted to start with B Angle B
I d like I did over here then be a
D is going to be congruent to triangle?
C d a kind of the same statement, but I'm going to use a triangle symbol instead of an angle symbol and of course
Use the sine angle side theorem
So we've got our triangles are ruined and that means that we can say that any
corresponding part of those two triangles are congruent themselves and
side bd of Triangle B
Ad and side see a of triangle CDa those are corresponding parts of those remote triangles
and they just also happen to be the diagonals of our trapezoid and they are congruent, so
CPCTC helps us prove that our diagonals of our asansol [Iza]. Trap. We are congruent that means
I'm done with my last example that means I'm double this video. [I'm] [just] true back go do your homework
you