 Hey everyone, welcome to tutor terrific today. I want to go over a neat geometry Investigation with you and that is in what it's called inscribed objects in a circle Whenever I inscribe an object in a circle, that means the entire object fits inside the circle And if it's a polygon that would mean all its vertices touch the circle So when something's inscribed that means it's completely inside the circle a very important Thing to understand are inscribed angles. I have an example of that right here Now notice that the center of the circle is not really involved in the situation When we are discussing inscribed objects Not usually and so here have an inscribed angle So that means its vertex is on the edge of the circle because it really only has one vertex That vertex is on the edge of the circle We usually stop the rays of the angle at the other points where it intersects the circle's edge also It inscribed angle does what is called intercept an arc on the opposite side of the circle So where it's two edges of its sides meet the circle We can look at an arc between the two of them and that arc would have a certain angle measure now We you might be familiar with what is called a central angle That is an angle from the center of the circle To an arc and it intercepts the arc as well Let's look at a central angle that intercepts this particular arc I'm gonna say this particular arc is 92 degrees Okay, just ballpark estimating and so the central angle by definition of central angles is also 92 degrees Okay, that's true What do you think the measure of this angle is? Do you think it's related? Do you think it's unrelated? Well, in fact, it is related. It's exactly half The size of the intercepted arc which in this case would make it 92 divided by 2 46 degrees All inscribed angles are exactly half of the intercepted arcs angle measure The funny thing is the interesting thing is as I can draw inscribed angles all over this circle If they intercept this same arc They are equal in degree measure to The other inscribed angles Here's another one if I draw from here That's also 46 degrees so you can basically move this point along the circle And if you leave the other two ends of the angle in the same position Your angles will all be 46 degrees Half of the arcs angle measure quite interesting. So that's What you could do or how you can understand inscribed angles Now let's look at something else that is very important that is inscribed in a circle quadrilaterals You can either call this as inscribed quadrilateral or a cyclic quadrilateral as it's called What's very interesting about these quadrilaterals is that opposite angles by label these one two three and four Opposite angles actually add up to 180 degrees in all cyclic quadrilaterals, and it's always Opposite angles so for this cyclic quadrilateral cyclic, excuse me measure angle one plus measure angle three equals 180 degrees Likewise the measure of angle four plus the measure of angle two also equals 180 degrees and As you might know that all quadrilaterals all four interior angles have to add to 360 Which this shows that they still do assuming that's true I'm not going to do the proof of this opposite angles being supplementary But there is a proof out there Again notice how the center is not involved. This is true for any inscribed quadrilateral. I'll draw another one Totally different than the first Opposite angles of this cyclic quadrilateral are also supplementary as are this one and this one Any inscribed quadrilateral has opposite angles supplementary. So now let's do a few example problems to be exact For inscribed angles in this first one here are more relatively easy problem. I want you to find both z and y Z is this arc right here. That's its measure In degrees and then y is the measure of this inscribed angle. Okay Let's start with y well As you can see y is inscribing the same arc this unnamed arc right here as this 28 degree inscribed angle is So that means y must be 28 Degrees. Yes, it's that simple two angles that are inscribed that intercept the same arc will have the same measure Now in order to find z we're going to need to do a little more work In order to uh move forward. I have to realize that z Plus this arc plus 131 has a special value. Does anybody know what it is? It's 360 degrees So if I could find out what this arc is And since I already know what this arc is z is the leftovers. So I take 360 minus this minus 131 and get z Perfect. Well, what is the measure of this arc? Well, it's intercepted by two Ininscribed angles that measure 28 degrees. So I would just multiply that by two to get to the arc degree measure 28 times two is 56 degrees So this arc is 56 degrees from here to here So z will then be 360 Degrees minus 56 degrees. Let's subtract this part and then minus 131 degrees If you pull out your calculator You will find that that is exactly 173 degrees All right, so we have our two answers For our two angles in this simple problem Now onto a more Challenging one that is um quite infamous for any of you who use the discovering geometry curriculum We have here two chords that intersect And the angle between them is 88 degrees We are given one arc On the circle it's 94 degrees and we are told to find f Which is a completely different arc And um seems unrelated and this seems impossible to first glance number one. I'm not going through the center This 88 degree angle is thus not a central angle. So I can't find this arc I have no information about this arc even though these are vertical angles I could find all of these angles in here, but none of them are central angles So they can't really be used to find f In some situations and some more challenging problems you actually have to construct Lines in the triangle or excuse me in the circle or whatever figure you have and I'm going to do that right now I'm going to construct this right here Now what this does is make some really interesting inscribed angles Okay, I want you to notice that this green line and this black line Are the edges of an inscribed angle to the arc Whose degree measure is known 94 degrees If you split that into two You get 47 degrees So this angle right here Is 47 degrees Okay Now if that angle is 47 degrees and this angle is 88 degrees I have a triangle here. Can I find the third angle? You betcha. What I have to do Is take 180 and subtract 88 And subtract 47 when you do that you get 45 degrees So this angle is 45 degrees Ah Here's the kicker Notice that this 45 degree angle ends here and here Which intercepts this arc Which is none other than f So all we have to do is double 45 degrees and we will find f 45 times 2 well, that's 90 So f equals 45 times 2 Which is 90 So that wasn't actually that bad Once you constructed this line and saw how it created two inscribed angles that would be very useful and connected All right, so here I have two cyclic quadrilateral problems for you This first one is a pretty basic one where you're given two angles in the quadrilateral and ask to find the other two Remember opposite angles in quadrilaterals that are inscribed in circles are supplementary Also remember that it has to have all the vertices touching the circle for this to work So let's find b Well b and 38 have to add up to 180 So b would be 180 degrees minus 38 degrees And if you do that subtraction you get 142 Very simple So b is 142 degrees now let's find c there's two ways we can find c We could do the standard 360 minus the other three angles Or we could make use of the cyclic quadrilaterals having opposite angles supplementary Which seems easier to me. We take 59 its opposite angle and subtract that From 180 To find its supplement Which would be c and that's 59 if you do that quick in your head. That's one less than 60 And so it'd be the result would be one more than 120 121 degrees It is that simple, but don't worry. I can make it harder This one is kind of the cream of the crop of both of these problems these both of these ideas Both these inscribed figures trying to find x and y notice how I said That this uh, it's a pair of sides is parallel Ah, okay, so that's going to help us find x because if you look at this at first glance you think there's no way There's no way I can do this. I don't have enough information I don't have uh two angles. Am I supposed to find One angle if I only have one angle in a cyclic quadrilateral Well, you can make use of those parallel sides If you look at this side that Connects the two angles And the parallel lines if you were to connect or extend them Just for a thought experiment You would notice that this line is like a transversal crossing parallel lines Now x and 95 in that case would be consecutive interior angles. They're on the same side of the transversal And they are interior. They're inside the parallel lines Consecutive interior angles are supplementary. It turns out which is interesting We thought opposite angles of supplementary will so our consecutive interior angles When two sides are parallel So to find x we can do 180 minus 95 Which would give us Exactly 85 degrees So x is 85 degrees Okay, great, but how in the world does that help us find y? Look at y. Why is this arc Outside the circle or at least that portion of the circle Well Look at x Look at the angle x Look at where it ends x is none other than an inscribed angle in fact for any cyclic quadrilateral All of its angles are inscribed angles quite useful Because that will allow me to find the other part of the circle Which the angle x intercepts that arc will be double 85 degrees So 85 times 2 Which you can do pretty much in your head. That's 170 because 80 times 2 is 160 So this arc Is 170 Okay, the remainder of the circle is Why? That arc So to find y we would just do 360 minus 170 And that my friends is 190 So y is 190 degrees So I just showed you how to use inscribed angles and cyclic quadrilaterals To solve some unique problems you might see in your geometry class or your math 2 course Good luck on those courses and enjoy. Thanks for watching guys. This is Falconator signing out