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

Lesson 8: Inscribed shapes problem solving# Inscribed shapes: angle subtended by diameter

One method using an inscribed angle intercepting a diameter of a circle.

## Want to join the conversation?

- At1:08, Sal says that angle SIE is supplementary to the 61 degree angle. This could only be true if LE is a straight line, but how can we tell this? Line LI and IE might be line segments going in slightly different directions. How do we know that line LE is a straight line without being explicitly told?(14 votes)
- During an entire year of Geometry, I never came across a problem where I couldn't assume lines to be straight if they appeared that way. So as far as I know, you can assume in Geometry that lines that appear straight are straight. You can't assume that they are parallel or perpendicular, or anything like that, but you can assume they are straight.

If I remember properly, I answered one of your questions a few days ago about assuming vertical angles? Well, this is the reason that you can assume vertical angles. If you assume the lines to be straight, then you can assume apparent vertical angles to actually be vertical angles.(21 votes)

- So when you are using surface area (2Ab+Pb times the Height of the prism) for a triangle, when you find the diameter... is it 90 degrees? or 180? or am I so completely lost on diameter of a triangle?(4 votes)
- I don't think he mentioned the diameter as much. He was probably working on the angles, since we're finding for the angle ISE. And he's not using the surface area either.... rather extending the triangle.(4 votes)

- At5:33, Sal said that arc SE measures 180 degrees. So, how do we get to know the measure of some other arc like arc LD in this circle?(Or just simply tell me that how do we get to know the measure of an arc)?(3 votes)
- Steps to find the measure of arc LD:

1. Since SE measures 180° because it is a straight angle, then ∠SLE and ∠SDE equal to 90° because they're inscribed angles that open up to a semicircle (180°), so you fill those angle measurements into triangles SLI and IDE.

2. Because vertical angles are congruent, ∠LIS and ∠DIE equal to 61°, so that leaves the remaining unknown angles in both triangles to be 29° (180°-61°-90°).

3. ∠LSD and ∠LED are inscribed angles that measure 29°, so their shared arc, LD, is twice their angles measures, so arc LD is 58°.(3 votes)

- hello, in the questions here they show or mention which points should be taken as the diameter, in my book it doesnt mention that in the question. how do i know what to take as my diameter? Eg: Points R(-2,1) , S(4,3) and T(10,-5) lie on the circumference of the circle. Find equation of the circle. How do i know which is my diameter?(2 votes)
- Figure out which two points if connected goes through the center of the circle and cuts it in half.(3 votes)

- At1:10, Sal says that angle SIE is a supplementary angle. How did Sal know that LE was a diameter?(2 votes)
- LE isn't a diameter... It's just a straight line segment and <SIE and <LIS add up to 180 degrees which makes them supplementary. A segment doesn't have to be the diameter to be straight. A diameter has to pass through the center of the circle to be a diameter. In this case that means that to be a diameter a line segment would have to pass through point O.(3 votes)

- At1:16how do we know that it is a straight angle?

What if its 179.99999, or 180.99999?(3 votes)- The people who make such questions make them in such a way to help you learn and practice. They don't make them to give you a wrong answer and frustrate you because you thought a 179.99999 degree angle was a 180.(1 vote)

- I know this was 2 years ago but I was wondering why would you subtract 1 after u do 180-61=120 at1:25why would u subtract 1 from 120? Instead of <I just being 120 because I dont see a reason to subtract 1 from 120 for the measure of angle I? Btw Im In the 9th grade so if u can explain it to where I can understand it I would really appreciate it Thanks! 😌 Have a wonderful Day and BE SAFE!(1 vote)
- Basically, he expanded 61 into 60 and 1 to make subtraction easier.(1 vote)

- I got a question for the diagram, since angle SIL subtends to the same arc as angle SEL, shouldn't angle SIL be the double of angle SEL?(1 vote)
- No, angle SIL is not a CENTRAL angle, as explained by Sal from before. Hope that helps! :-)

@Minigeek0105, I edited it.(1 vote)

- how would I do this if angle LIS was unknown to me?(1 vote)
- so all inscribed angles are half of the arc it intercepts?(1 vote)
- Yes. It is a theorem called
*the inscribed angle theorem*. It says that the measure of an inscribed angle is half of the central angle that subtends the same arc. Hope this helps! 😊(1 vote)

## Video transcript

- [Voiceover] "On circle O
below", so this is circle O, "segment SE is a diameter." So this is SE, let me color that in. So this is a diameter, that's
what they're telling us. And they say, "What is the measure of "angle ISE?" So what we care about, the measure of angle ISE. I, S, E. So we're trying to figure out this angle right over there. Now like always, I encourage
you to pause the video and see if you can work
through it yourself. So there's a bunch of
ways that we can actually tackle this problem. The first one that jumps out at me is there's a bunch of
triangles here, and we can use the fact that the
angles, interior angles of a triangle add up to 180 degrees. So we could look at this
triangle right over here. And so we know one of the angles already. We know this angle has a
measurement of 27 degrees. If we could figure out
this angle right over here, this would be angle
SIE, then if we know two interior angles of a triangle, we can figure out the third. And this one, SIE, we can figure it out because it's supplementary
to this 61-degree angle. So this angle right over
here is just going to be this angle plus the 61-degree angle is going to be equal to 180 degrees, because they are supplementary. Or we could say that this angle over here is going to be 180 minus 61. So what is that going to be? 180 minus 60 would be 120, and then minus one gives us 119. 119 degrees. And so this angle that we
are trying to figure out, this angle plus the 119 degrees, plus the 27 degrees, is going to be equal to 180 degrees. Or we could say this angle is going to be 180 minus 119, minus 27. Which is going to be equal to, so let's see, 180 minus 119 is 61. And then 61 minus 27 is going to be 34. So there you have it. The measure of angle ISE is 34 degrees. Now I mentioned that
there is multiple ways that we could figure this out. Let me do it one more way. So let me unwind everything that I just wrote. We already figured out the answer, but I want to show you that there's multiple ways that we can tackle this. So ISE is still the thing that we want to figure out. Another way that we could approach it is we know we have some angles inscribed angles on this circle, and we know that if an inscribed angle intercepts a diameter,
then it's going to be a right angle. It's going to be a 90 degree angle. So this angle right over here is a 90 degree angle. And we can use that information to figure out this angle, and we can also use that information if we look at this triangle, we could use 90 plus 61 plus this angle is going to be equal to 180 degrees. So this angle right over here, another way to think about it, it's
going to be 180 minus 90 minus 61. Which is equal to, 180 minus 90 is 90, minus 61 is 29 degrees. So this one right over here is 29 degrees. And then we could look
at this larger triangle. To figure out this entire angle. If we know this entire
angle, you subtract 29 then you figure out angle ISE. And so this large, or what I've depicted, this kind of magenta
measure right over here of that angle, plus 90 degrees, plus 27 degrees, is
going to be equal to 180 because they're the interior
angles of triangle SLE. So this angle right
over here is going to be 180 minus 61 minus 27. Sorry, not minus 61, minus 90. It's 180 minus 90, minus 27 is going to give us this
angle right over here because the three angles add up to 180. So minus 90, minus 27. Which is equal to, so 180 minus 90 is 90. 90 minus 27 is 63. 63 degrees. So this large one over here is 63 degrees, and then the smaller one is 29 degrees. And so angle ISE, which
we set out to figure out, is going to be 63 degrees
minus the 29 degrees. So 63 minus 29 is once again equal to 34 degrees. So the way I did it just
now, a little bit harder. It really depends what jumps out at you. The first way I tackled it,
it does seem a little bit easier, a little bit clearer. But it's good to see
these different things. And at least here, we used this idea of an inscribed angle that
intercepts a diameter. And if you say, "Hey, how do we know?" Well, we've proven in other videos, but it comes straight out of the idea that the measure of an inscribed angle is going to be half of
the measure of the arc that it intercepts. And notice, it's intercepting an arc that has a measure of 180 degrees. It's intercepting an arc that has a measure of 180 degrees. And so this angle's
going to be half of that since it's an inscribed
angle, not a central angle. So it's going to be a 90 degree angle.