Circles

# Inscribed angles

## Inscribed angle theorem proof

Proving that an inscribed angle is half of a central angle that subtends the same arc.
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## Inscribed angle theorem proof

Discussion and questions for this video
At 0:25, is there a special meaning si has? Oh, and theta too. (2:01.)
psi and theta are just like using x in algebra they are just variables
what is sy and data?
PSY is a Korean music artist known for his songs "Gangnam Style" and "Gentleman" and data is a term for information used usually in context with computers and the internet.

But you are probobly wondering about PSI (Ψ) and THETA (θ) which are the 23 and eighth letters in the greek alphabet, respectively. Sal sometimes uses them as variables, instead of x or y or whatever.
At 8:22 how do you know psi 1 is = to 1/2 beta 1, and same for psi 2 and beta 2?
At 6:00 minutes of this video Sal proves psi = 1/2 beta when one ray is a diameter, in the example you're asking about he creates a diameter so he can use this result to know Psi 1 = 1/2 beta 1 and the same for psi 2 and beta 2.
at 00:38 you mentioned si for the first time. What does si signify?
Psi (ψ) (not si) is a Greek letter. It's just a variable, like theta (θ). He's using it because there are two important angles so he needs a different letter for each.
Could you do a problem like this with different variables?
Like, instead of Psi and Theta, could you use x and y?
Yes, you can use whichever variables you like. But x and y are commonly used to indicate lengths, and psi and theta are commonly used to indicate angles, so that's what Sal has done.
At 0:11 and 0:45, Sal says "vertex", what does that mean?
The Vertex is the point where the two line segments meet. So at 00:11 when he points to the vertex he is staying, the place where the two line segments meet, which is on the circumference of the circle
I had a question about theta. If someone draws the symbol theta would you draw the line that goes in the center only as long as the radus or to the diameter? Thank you very much.
Theta is simply a variable that he uses to describe one of the two types of angles used in the above video: central angles. The symbol theta has no meaning specific to the concept discussed in the above video; like Psi, which he uses to indicate the inscribed angles, the Theta symbol is used to represent something, therefore it is not how the symbol is drawn that matters--it is what the symbol represents that matters.
Can the rays of Si be outside of the circle and not inside?
The answer is yes. The rays forming si can intersect outside the circle. In this case, the the angle will sustend two arcs on the circle. The measure of angle si will then be 1/2(arc2 - arc1). Arc2 being the larger arc and Arc1 being the smaller arc.
1 Comment
Wait, in that third situation, what garuntee do you have that si 2 = theta 2? There are three triangles in that part of the video (at 11:25) and in the other parts, then there are two triangles? It doesn't really make much sense.
he's taking from the first example- if one of the chords of the angle is the diameter then we already know that psi= 1/2theta, we already proved that. so using that we know that psi2=½(theta2) and we can and since the huge angle we made also has one of the chords on the diameter, we can state that psi1+psi2=½(theta1+theta2). Since we know that psi2=1/2theta1, we can plug it in and now we have psi1+1/2theta1=½theta1 +½ theta2 and the term 1/2theta1 cancels from both sides and you are left with psi1=1/2theta1, which are the two measurements we were looking for.Hope this helps.
At 10:05 What is "si"?
Actually, it is Psi, don't do Psy because he is a K-pop star.
Psi is a Greek letter used to represent an unknown value of a angle formed by 2 chords which have a common endpoint. (A inscribed angle)
I don't get it... why not just use X or Z?
Because X and Z are for algebra. psi (but the p is silent) and theta. They're just variable traditionally used for angles( they are Greek letters)
I never seen Psi used as a variable like that. Is there a convention to use it for inscribed angles?

Psi reminds me of the mind. It brings up words like psychic and psychology. I think Cupid had a girlfriend, name Psyche. Her name meant soul.
The letter _psi_ (uppercase Ψ, lowercase ψ) is commonly used in physics, to present an understanding of functions in quantum mechanics. You are correct to state that it is used in words like Psychology. Although, I'm uncertain of the usage in Cupid's *wife* (not girlfriend). It could be that it refers to the mind( and soul, as you state).
Theta and Psi are letters in the Greek alphabet. They are commonly used to represent angles, the same way letters like a, b, c, or x, y, or z are commonly used as variables to represent unknowns in algebraic expressions. He could have used any letter to represent the angles. Hope this helps. Good Luck.
what are planes
A plane is a flat surface with a thickness of 0. Any three points not on the same line will make a plane.
what's psi annd what are its uses ?
Although the definition for p.s.i above is correct, what is used in the video is Psi, a Greek letter. It is used to represent an unknown angle, like a more traditional x or y.
Geometry can be taken in 8th grade, I don't think any 7th graders take it but maybe, and of course any high schooler can take it, 9-12th, but its usually 10th grade (:
how does he know psi 1 equal 1/2 theta 1 at 8:15 in the vidio? i didn't undrstand the proof. i don't think i saw a proof. like wise for scy 2 and thaita 2 at 8:20
Sal wasnt really proving anything in particular. All he wanted to show was that the centre neednt be within the arc being subtended.
And he just did that using stuff that he had taught before
He clearly shows that an inscribed angle is one half of the central angle that subtends the same arc but does that mean that it is one half of the arc that it subtends as well?For example if an arc is 90 degrees does that mean that the inscribed angle subtended by that arc is 45 degrees?
I think that when you're measuring an arc in degrees, you're basically finding the central angle. So if I understood both questions correctly, it's both yes.
Are Psi and Theta used as placeholder for a value?
Yes. Just like x and y, they stand in for a value you don't know at the moment or else as a variable that might change. θ and ψ are nearly always used to represent angles, while x and y are used for points, lines, and curves.

For reference sake, just as x is the standard variable to use for an unknown, and if you need a second one it is standard to use y; with angles it is standard to use θ as the first unknown angle's variable and ψ for the second unknown angle. However, some people (myself included) tend to use α and β after having used θ.
But, really, it is all just arbitrary placeholders. You can call it anything you like, but math teachers tend to like for you to stick to traditional variables.
1 Comment
Why doesn't the question page show up? I can't see the answer to my own question!
Go to your profile and click on discussion to see questions you've asked and answers you've given. There's a lot of people on here, so it can be hard to find yourself without that.
Since si is an inscribed circle and theta is the central angle that subtends the same arc as si, shouldn't theta be 2 times si and not 1/2 of si?
What would happen if the vertex of the inscribed angle is in between its endpoints?
why psi and theta instead of a or b? Do they stand for something or are they just variables?
a and b and other alphabetical variables are traditionally used in algebra.
For geometry, Greek alphabets like psi and theta are used. They are just variables like a and b.
What does Sal mean by the term, "subtended"?

Subtend
/səbˈtend/
• (of a line, arc, or figure) form (an angle) at a particular point when straight lines from its extremities are joined at that point.
so psi and theta are just variables like x and y?
then why is it used over x and y in trigonometry/geometry if they are essentially the same?
psi and theta are used to represent angles. x and y are used in geometry to represent length of sides. That way if you see psi or theta you know they are talking about the angle.
What does Sai or Si, whatever you call it, mean?
do you mean psi?
if so, its a Greek letter used to denote angles
is vector geometry?
I don't think so. https://goo.gl/3P5XdQ
You have explained this much better than my teacher! Can you please make more geometry videos. Especially using the Common Core.
What does Subtended mean at 1:32
It's the amount of arc that's 'cut out' by the two lines extended from the angle we're talking about. - that should be clear from the vid @1:32
Is Si a mathematical expression like theta or is it representing a number?
It is spelt Psi. See http://en.wikipedia.org/wiki/Psi_(letter). It is a greek letter, like alpha, beta, gamma, pi. For a full list of greek letters and how they're used in mathematics and science see http://en.wikipedia.org/wiki/Greek_letters_used_in_mathematics,_science,_and_engineering. Sal could've picked another letter if he wanted. Perhaps he wanted to have a change for this video.
Should we always use Si and Theta (<-- don't know if i spelled it right) or are they merely just being used as variables?
They are simply just used as variables. You can really use any shape or letter you would like. Even smiley faces could work in place. :)
When my teacher does these in class, she measures the outsides of an arc in degrees. Aren't we supposed to use regular measurements(Metric/Customary)?
You can use measurements like that if you are provided units in the first place. Usually you should use the units provided in the problem. If there aren't any units use radians or degrees. Arc length is usually always measured in radians which are convertible to degrees. Hope that helps!
If the interior angles of a circle add up to 360 degrees, then what about an ellipse? Would the angles of an ellipse add up to 360 degrees as well?
Yes. You know how all squares are rectangles, but not all rectangles are squares? The same is true for circles. All circles are ellipses, but not all ellipses are circles.
Since a circle is just a special ellipse, then if a circle = 360, then so does an ellipse.
1 Comment
I have always wondered is it Theta or Beta?
Theta and Beta are just traditionally used variables for angles.
College Board Sat Official Practice Test Question: #8. Square RSTU is inscribe in the circle. What is the degree measure of arc ST?
can i get a credit for watching these video taking notes
what does si mean or stand for? i dont get it:(
psi is a greek character, he's just using it to denote the angle.
They say triangles are the strongest shape, but wouldn't circles be because they have no weak points to break under? Then we could have bridges and towers with circular supports like those ancient arches.
Well, this seems like more of a fantasy. Don't argue with Euclid or any geometry expert who knows more than you about geometry who says that triangles are the strongest shapes. Circles are not the strongest.
HEY guys(gals) I have a question... what is the meaning of si(the pitchfork looking thing) and beta? (the circle with the line through it?)
Psi and theta are Greek letters. They are used many times as variables for unknown angles.
Is si/psi spelled si or psi? I've seen both spellings this video.
It is spelled "Psi" and the sign Ψ is the 23 letter of the greek alphabet which means Ps
What is subtending?
Subtending of an angle is the angle created when two lines intersect in a circle at the center.
Could you have used the Exterior Angle Theorem, where it says that an exterior angle of a triangle is equal to the sum of its two remote interior angles, in the first example and skipped the supplementary step?
Yep. I'm assuming he's keeping things basic to help keep the topic understandable, not using too much big terminology at once, I'd assume.
What is "theda"?
It's "theta". It is a Greek letter that is used mostly to represent unknown angles.
how to find lettered angles in a circle?
The lettered angles are just variables, like x or y. Think nothing differently of them then you would if you were dealing with a regular a,b,c variable.
is SI a greek character??
yes and it's psi not si. =)
Why does Sal use psi? What is the purpose of Psi? I dont use psi and i also didnt know what it was.
It is standard practice in mathematics to use Greek letters to represent angles. Among the more commonly used are θ, ψ, φ , α, β, γ (Theta, psi, phi, alpha, beta, and gamma). Sometimes you'll see people use omega (ω).
It would be a good idea to learn the Greek alphabet. You don't necessarily need to know the names of all the letters, but just be able to recognize and draw them (though I would recommend learning the names of the more commonly used Greek letters)
okay this is a good video but my school isn't learning si or data so now I'm even more confused what do they stand for?
Psi and Theta are just variables like "x" and "y" that stand for angles. They are greek letters.
what does subtending mean?
Subtending an arc means that the arms of the angle are separating a part of the circle. They are creating a new arc of the circle.

It basically is saying that there are two (or more) angles whose arms end in the same place.
how does sal gets 1/2 at 5:50 - 5:56
He divided both sides of the equation by 2.
i.e. if 2 pens are worth 1 dollar, then 1 pen is worth 1/2 a dollar.
What is the biggest and smallest possible angle
Technically, the smallest angle could be 0 degrees, and the largest, 360. But, with both of those the angle is practically either ending, or just beginning, and is really non-existent. So, the smallest angle possible would be 1 degree large (or a decimal value of that, such as 0.000001), and the largest angle would be 359 (or again, a decimal value, such as 359.99999). Any smaller or larger (depending on what we are considering) and the angle starts all over again, or becomes nothing at all. Does that make sense?
find RVT if <RST is a central angle of circle S.
S = 118
RST is because if the circle is called circle S, than <S = <RST.
what are these ideas useful for?
Does 'subtend' just mean that it goes through an arc of the circle?
am i spelling this right?: si and data
its psi; theta
Why is it the only actual number I see in this entire video is 180? Don't the other angles have to be numbers?

I've only watched this video five times now. Having trouble conceptualizing this.
its all unknowns- the only thing you know for sure is that the angle of a straight line is 180. the ffigures are only there to help, they dont show the actual angle, therefore they are represented by variables. in geometry, greek letters are used as variables just as english letters are used in algebra. Yes, the other angles have to be numbers, but we dont know exactly what numbers they are.
Whhat does he mean when he says, when we have two sides being equal (isosceles) then the base is equal? What is the side and what is the base?
Could someone tell me what subtend means? Thanks.
Subtend means "intersect" or "encompass." For example, at about 2:00, the purple arc is subtended by psi and theta, that is, it is "inside" the angle.
What does 'si' (that cactus-shaped thing that Sal always uses) mean? It's the first time I've ever heard anyone use it in my whole life.

-mdanivas, a.k.a. Pants
This is Theta: ϴ and this is psi: ψ and they just are GrΣΣk letters used to mean unknown in Geometry.
How do you find out Si and Theta?
I'm kind of confused. Can someone please explain this to me in a simpler way?
At 12:37, isn't the Theta 1 and Theta 2 a 90 degree angle? Cause Salman have drawn the angle as similar as 90 degree.
no,you can't say that thetas is 90 degrees just by observing and not measuring them
At 3:34, the Isosceles triangle that it says isn't really true, because the center isn't accurate enough.
If you measured it out, including the exact placement of the center along the diameter, then yes, you would probably find that the center is off by a little. However, because a diagram is a *representation* of something, some liberties can, and are, often taken with it. For instance, if you opened your math book and found a diagram of the base of the Statue Of Liberty along with it's measurements, you (probably) wouldn't think "Oh, well these are wrong. It says here that the height of the base is 89 feet, but on this diagram, it looks like it's only 2 inches.". That's because you know it's a diagram, and it's only showing you a general idea of what it might look like. Some diagrams are often equipped with a note, saying "diagram not drawn to scale", and that is the case with both the diagram of the Statue Of Liberty's Pedestal, and the diagram of the inscribed Isosceles triangle.
how would any non- physicist use this in their life?
About 7:00, he said that the notation won't apply to all inscribed angles. Is there a special case where the notation doesn't apply?
Thank you Sal. You are an excellent math instructor and this site is beyond helpful.
However, you tend to use a lot of algebra. I find it would be more helpful to use simple arithmetic.
at 8.10 Dont really understand how you can assume psi1 is equal to theta1. Isnt this a circular argument where you are assuming the conclusion for this particular repetition of the proof?
What is Si/Psi?
A greek letter used to note the inscribed angle...and is it psi
at 2:01 what does si has mean?
Psi (not si) is a letter in Greek alphabet (Ψ). Mathematicians use Greek letters to write angles. They often use other letters : α(alpha), β(beta), γ(gamma), θ(theta)
How would you solve PSQ=3y-15 and PRQ=2y +25?
The problem is: Find m<PSQ if M<PSQ=3y-15 and m<PRQ= 2y+25
< = angle sign
A circle with an area of 49π square centimeters is inscribed (tightly inside of) a square. What is the area of the square?
could u tell me the proof of the inscirbed amgle theroem, case 2?
When the angles form a triangle or a quadrilateral, how would you find the missing angles of the polygon inscribed in the circle?
what is the angle called when it is neither central nor inscribed? and what is the formula for it. does the arc equal the angle or is it the angle half of the arc? help!
would the central angle also show the measure of the arc (number of degrees)?
Only thing to do in exercise: blue angle divided by 2 = orange angle and orange angle times 2 = blue angle. Do that for all the 3 exercise on this and you get 100% :) logic
what does si represent