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### Course: High school geometry > Unit 8

Lesson 8: Inscribed shapes problem solving# Inscribed shapes: find inscribed angle

Using either the inscribed angle theorem or the fact that two inscribed angles that intercept the same arc must be congruent.

## Want to join the conversation?

- Hello Sal, I was wondering how does the length of Arc CD correspond to the measure of angle DEG, when the EC doesn't pass through the center of the circle?

It wouldn't be a central angle.(19 votes)- When he says the angle of the arc, he means the angle of the arc as measured from point O, which according to previous videos should be twice the angle.(7 votes)

- Sal proofed that the measure of an inscribed angle which intercepts the same arc as a CENTRAL ANGLE is going to be half the measure of that of the CENTRAL ANGLE. In this video he states, that an inscribed angle also has half the measure of the ARC, it intercepts. When did he proofed that? Did I miss a video. In the previous videos or articles, he only proofed it vor CENTRAL ANGLES, not for ARCS, right?(18 votes)
- I think you are right. However, the measure of the ARC in this case is not the length of the ARC as a portion of the circumference, but rather its measure as a portion of the 360 degrees that make up the entire circle. In this sense, the ARC and the CENTRAL ANGLE are the same thing.(10 votes)

- The highlight of this videoー "two inscribed angles that intercept the same arc must be congruent."(12 votes)
- 1:25How are two inscribed angles that are subtended by the same arc equal to each other?(5 votes)
- For any given arc, there can be any number of inscribed angles that subtend it, but only one central angle will subtend that same arc. Since the inscribed angle theorem tells us that any inscribed angle will be exactly half the measure of the central angle that subtends its arc, it follows that all inscribed angles sharing that arc will be half the measure of the same central angle. Therefore, the inscribed angles must all be congruent. Hope this helps!(10 votes)

- from where do we get the theorem that an inscribed angle will be half the measure of that arc which it intercepts? please help.(8 votes)
- Are intercepts and intersects the same?(3 votes)
- Intercepts is a noun to describe where the graph of a function or equation crosses the x or y axis. Intersects is a verb to describe where two or more lines meet, it could but does not have to be on the x or y axis, it could be anywhere on the graph.(5 votes)

- very hard to understand 100 degree thing and cannot get it after reading to many comments(4 votes)
- but the angle should be a central angle to apply the rule of the inscribed angles ; that one is not central since the centre is o not g(2 votes)
- That would be true if we were looking for the Measure of Arc CD, because the angle would have to originate at O.

BUT we don't have to do that here: because as Sal explains, the two angles we are comparing are both INSCRIBED angles. And an inscribed angle is ALWAYS half of the measure of an Arc. Which means that all inscribed angles of the same arc are going to be the same.

Since angle CFD is an inscribed angle of Arc CD, and angle CED is ALSO an inscribed angle of Arc CD, the two angles are equal.

Another way to think about it: you can use CFD to solve for COD. Since COD is going to be the Measure of Arc CD, and CFD is an inscribed angle, we know that CFD is going to be half of COD. So COD would HAVE to be 100degrees. And then we can turn it around, and see that CED is an inscribed angle of COD - and since an inscribed angle of COD is half of COD, we know that CED has to be 50degrees.(4 votes)

- how is arc CD twice angle CED AT0:44??

Please answer ,i am so confused.(3 votes) - what is an inscribed angle? definition wise, how do i know if an angle is an inscribed angle(2 votes)
- An angle is inscribed in a circle when it's vertex is on the circumference and both its rays fall within the circle.(2 votes)

## Video transcript

- So what I would like you to do is see if you can figure out
the measure of angle DEG here. So try to figure out the
measure of this angle. I encourage you to pause the video now and try it on your own. All right, now let's work
through this together and the key realization
here is to think about this angle, it is an inscribed angle, we see it's vertexes sitting
on the circle itself. And then think about the
arc that it intercepts. And we see, we see that it intercepts, so let me draw these
two sides of the angle, we see that it intercepts arc CD. It intercepts arc CD. And so the measure of this angle, since it's an inscribed angle, is going to be half the measure of arc CD. So if we could figure out
the measure of arc CD, then we're going to be in good shape. Because if we figure out
the measure of arc CD, then we take half of that and we'll figure out what we care about. Well, what you might notice,
is that there's another inscribed angle that
also intercepts arc CD. We have this angle right over here. It also intercepts arc CD. So you could call this angle C ... Whoops. You could call this angle CFD. This also intercepts the same arc. So there's two ways you
could think about it. Two inscribed angles that
intercept the same arc are going to have the same angle measure so just off of that you could
say that this is going to be, that these two angles
have the same measure, so you could say this is
going to be 50 degrees. Or you could go, you could actually solve what the measure of arc CD is. It's going to be twice the
measure of the inscribed angle that intercepts it. So the measure of arc CD
is going to be 100 degrees. Twice the 50 degrees. And then you use that and you say, Well, if the measure of
that arc is 100 degrees, then an inscribed angle that intercepts it is going to have half its measure, it's going to be 50 degrees. So either way we get to 50 degrees.