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### Course: High school geometry (staging) > Unit 9

Lesson 8: Inscribed shapes problem solving# Solving inscribed quadrilaterals

Example showing supplementary opposite angles in inscribed quadrilateral.

## Want to join the conversation?

- I don't quite get how khan got 90 degrees for the small arc.(14 votes)
- We know that the measure of an arc is DOUBLE the measure of the Inscribed Angle. In this case, the
**angle WIL**is inscribed by the blue arc. The measure given for this angle is 45*degrees*.

So, the measure of the blue arc is 45*2=90*degrees*.

I hope you got that now.(16 votes)

- Okay, this is basically turning my mind to mush. How are the angles always supplementary? Where did he get 90 degrees from? I'm confused, please help me.(8 votes)
- Angles for inscribed quadrilaterals are always supplementary. It's one of those weird facts of math that they don't normally explain, like why you divide before adding, etc. If you do some research, you can likely find some proof for why inscribed quadrilaterals are always supplementary, but it's not something normally covered in math courses. Sal got the 90 degrees by multiplying 45 by 2. This is because of one of the theorems that states that inscribed angles are 1/2 of the degree measure of the arc they intercept. Hope this helped.(1 vote)

- You can just do 180-45 because the opposite angles of a quadrilateral have a sum of 180 degrees.(5 votes)
- I still not understanding the math its really stressing to learn about the subject but am getting there(5 votes)
- is the opposite of angle WDL a valid inscribed angle(2 votes)
- WIL, ILD, LDW and DWI are all inscribed angles

An inscribed angle is the angle formed from the intersection of two chords, and a chord is a line segment that has each end point on the side of the circle somewhere.

So there are 4 chords, WI, IL, LD and DW and each place they intersect forms an inscribed angle.

I assume by opposite you mean WIL, but all angles there are inscribed angles.

Let me know if this did not help.(3 votes)

- at4:18in video he talks about the sums being supplementary. are they Always supplementary? how can you tell if they are or not?(2 votes)
- From what Sal said in the video after that point, he implies that they are always that way and he is going to prove it in a later video.(3 votes)

- Wait, the angle is not inscribed? What?(3 votes)
- At what point exactly in the video you're getting confused? Let us know so we can help!(1 vote)

- 4:18I thought triangles could only add up to 180 not 289(2 votes)
- It is not triangle there it's an quadrilateral(2 votes)

- From1:53to2:07, why did the teacher mention that arc (WL) was equal to 90 degrees? Please clarify.(2 votes)
- Explanation starts at1:22but basically since the inscribed angle is 45 degrees the arc measure must be 90 degrees because 45 times 2 = 90 and inscribed angles are half of the measure of the arc (bad explanation ik)(1 vote)

- Interesting question about the relationship between arc measure and the angle that intercepts that arc.

In the video, if you want to find the arc that angle O intercepts, which is 180 degrees, you will get 360 degrees. And clearly that arc is half of the circle and isn’t 360 degrees. So how does it works.(1 vote)- Angle O is 180 degrees, but it is a central angle, not an inscribed angle. The measure of the central angle is equal to the measure of its corresponding arc. I think that you thought you have to double the measure of a central angle to find its corresponding arc, but this is only for inscribed angles. Kinda late but...🤷♂️(3 votes)

## Video transcript

- [Voiceover] What I wanna
do in this video is see if we can find the measure of angle D, if we could find the measure of angle D and like always, pause this video, and see if you can figure it out. And I'll give you a little bit of a hint. It'll involve thinking about
how an inscribed angle relates to the corresponding to
the measure of the arc that it intercepts. So, think about it like that. All right, so, let's work
on this a little bit. So, what do we know, what do we know? Well, angle D, angle D intercepts an arc. It intercepts this fairly
large arc that I'm going to highlight right now
in this purple color. So, it intercepts that arc. We don't know the measure of that arc or at least we don't know
the measure of that arc yet. If we did know the measure of this arc that I'm highlighting, then
we know that the measure of angle D would just be
half that because the measure of an inscribed angle is half the measure of the arc that it intercepts. We've seen that multiple times. So, if we knew the measure of this arc, we would be able to figure out what the measure of angle D is. But we do know, we don't
know the measure of that arc, but we do know the measure of another arc. We do know the measure of the
arc that completes the circle. So, we do know the measure of this arc. You might be saying,
hey wait, how do we know that measure, it's not labeled. Well, the reason why we
know the measure of this arc that I've just highlighted
in this teal color is because the inscribed
angle that intercepts it, they gave us the information, they said this is a 45 degree angle. So, this is a 45 degree angle. Then this over here is a
90 degree, 90 degree arc. The measure of this arc is 90 degrees. The measure of arc, I guess you could say this is the measure of arc,
I'ma write it this way. The measure of arc, WL. WL is equal to 90 degrees, it's twice that, the inscribed
angle that intercepts it. Now, why is that helpful? Well, if you go all the
way around the circle, you're 360 degrees. So, this purple arc that we
cared about, that we said hey, if we could figure out
the measure of that, we're gonna be able to figure
out the measure of angle D. That plus arc, WL, they are going to add up to 360 degrees. Let me write that down. So, the measure of arc, let's see, and this is going to be a
major arc right over here. This is so LIW, the measure of arc LIW plus the measure of arc WL plus the measure of arc WL plus this right over here. That's going to be equal to 360 degrees. This is going to be equal to 360 degrees. Now, we already know
that this is 90 degrees. We already know WL is 90 degrees. So, if you subtract 90
degrees from both sides, you get that the measure of
this large arc right over here, measure of arc LIW is going to be equal to 270 degrees. I just took 300, I went all
the way around the circle, I subtracted out this 90 degrees and I'm left with 270 degrees. So, let me write that down. This is the measure of this
arc in purple is 270 degrees. And now, we can figure out
the measure of angle D. It's an inscribed angle
that intercepts that arc so it's going to have half the measure, the angle's going to
have half the measure. So, half of 270 is 135 degrees and we're done, and you might notice
something interesting, that if you add 135
degrees plus 45 degrees, that they add up to 180 degrees. So, it looks like at least for
this case that these angles, these opposite angles of
this inscribed quadrilateral, it looks like they are supplementary. So, an interesting question is are they always going to be supplementary? If you have a quadrilateral,
an arbitrary quadrilateral inscribed in a circle,
so each of the vertices of the quadrilateral sit on the circle. If you have that, are opposite
angles of that quadrilateral, are they always supplementary? Do they always add up to 180 degrees? So, I encourage you to think
about that and even prove it if you get a chance, and
the proof is very close to what we just did here. In order to prove it, you
would just have to do it with more general numbers like, you know, instead of saying 45 degrees,
you could call this X, and then you would want to
prove that this right over here would have to be 180 minus X. So, I encourage you to do that on your own but I'm gonna do it in a video as well so you can check if our
reasoning is similar.