High school geometry
Sal uses the inscribed angle theorem and some algebra to prove that opposite angles of an inscribed quadrilateral are supplementary.
Want to join the conversation?
- at1:32, Sal says that in previous videos, it is shown that the measure of an inscribed angle is half the measure of the arc it intercepts. I can't find any video on Khan Academy showing this? All I can find are videos proving that the measure of an inscribed angle is half the measure of a central angle that subtends the same arc, which seems related to, but not quite the same as what he is getting at here. Could anyone point me in the direction of the video/ explain the connection between these two ideas?(29 votes)
- "The measure of an arc" means "the measure of the central angle that subtends that arc". It is just another way of phrasing the same thing. By default, if it is not specified, you should assume a given measure of an arc is the measure of the central angle that subtends that arc.
Also here is the lesson regarding inscribed angles and central angles, in case anyone isn't familiar with the proof behind it all:
- Does this work even if there is no circle? That is, all quadrilaterals have this property that the opposite angles are supplementary? I'm guessing not...and that's because the four vertices have to be able to form a circle. Is this right?(2 votes)
- You are getting rules mixed up, some quadrilaterals have opposite angles supplementary (isosceles trapezoids, squares and rectangles because they are all 90 and many trapeziums). For parallelograms, opposite angles are congruent and consecutive angles are supplementary.
So it works when it forms a circle. If you start with a parallelogram that is not a rectangle or rhombus, you could not draw a circle that has all 4 vertices on it.(5 votes)
- What theorem is this called (wanted to know for proofs)?(2 votes)
- At3:24, why does Sal do it as 1/2(360-2x)?(1 vote)
- The inscribed angle theorem which was shown in the video before this one. An inscribed angle ( one that touches the circle) is related to the angle it subtends by a scale factor of two, If I know the angle, I double it to get the arc (which he did at the beginning to get 2x), and if I know the arc, I cut it in half to get the angle which is where the expression you asked about comes from. I assume you are okay with the 360-2x to be all of the circle not part of the 2x.(3 votes)
- When we say "the measure of the arc", are we talking about the angle?(1 vote)
- The 'measure of an arc' or arc measure is equal to the arc length divided by the radius (s/r = 2x degrees). An arc measure is an angle the arc makes at the center of a circle, whereas the arc length is the span along the arc. In other words, an arc measure is equal to the central angle that subtends the arc.(2 votes)
- Is there a relationship between the length of the chord between the two endpoints of the arc and the value of the angle(with vertex at the center of the circle) ? If so, what would the graph look like(input: length of chord, output: Value of angle)? I am aware that this is not a function, but am still interested in the graph. Thanks in advance!(1 vote)
- I believe this isn't sufficient to list a relationship (since for a larger circle, the chord will also increase in length). You will need the radius of the circle as well. After that, it's just the formula of the arc measure.(1 vote)
- Is the measure of the arc subtended by the rays that form angle x the same as the measure of the central angle? Aren't they equal only if the radius is 1?(0 votes)
- No, regardless of the radius, the measure of the inscribed angle (angle with vertex on the circle) will be half of the central angle (angle with vertex at the center of the circle) that is formed by the same arc. All circles are similar and the radius will not necessarily change anything as far as simple angles go.(1 vote)
- Why do is the measure of x half of the arc measure?(0 votes)
- [Voiceover] So I have a arbitrary inscribed quadrilateral in this circle and what I wanna prove is that for any inscribed quadrilateral, that opposite angles are supplementary. So when I say they're supplementary, the measure of this angle plus the measure of this angle need to be 180 degrees. The measure of this angle plus the measure of this angle need to be 180 degrees. And the way I'm gonna prove it is we're gonna assume that this, the measure of this angle right over here, that this is x degrees. And so from that, if we can prove that the measure of this opposite angle is 180 minus x degrees, then we've proven that opposite angles for an arbitrary quadrilateral that's inscribed in a circle are supplementary, 'cause if this is 180 minus x, 180 minus x plus x is going to be 180 degrees. So I encourage you to pause the video and see if you can do that proof and I'll give you a little bit of a hint. It's going to involve the measure of the arcs that the various angles intercept. So let's think about it a little bit. This angle that has a measure of x degrees, it intercepts this arc, so we see one side of the angle goes and intercepts the circle there. The other side right over there. And so the arc that it intercepts, I am highlighting in yellow. I am highlighting it in yellow. Trying to color it in, so there you go. Not a great job at coloring it in, but you get the point. That's the arc that it intercepts and we've already learned in previous videos that the relationship between an inscribed angle, the vertex of this angle sits on the circle, the relationship between and inscribed angle and the measure of the arc that it intercepts is that the measure of the inscribed angle is half the measure of the arc that it intercepts. So if this angle measure is x degrees, then the measure of this arc is going to be 2x, 2x degrees. All right, well that's kind of interesting, but let's keep going. If the measure of that arc is 2x degrees, what is the measure of this arc right over here? The arc that completes the circle. Well, if you go all the way around the circle, that's 360 degrees. So this blue arc that I'm showing you right now, that's going to have a measure of 360 minus 2x degrees. 360's all the way around. The blue is all the way around minus the yellow arc. What you have left over if you subtract out the yellow arc is you have this blue arc. Now, what's the angle that intercepts this blue arc? What's the inscribed angle that intercepts this blue arc right over here? Well, it's this angle. It's the angle that we wanted to figure out in terms of x. Wow, I'm having trouble changing colors. It is that angle right over there. Notice, the two sides of this angle, they intercept, this angle intercepts that arc. So, once again, the measure of an inscribed angle is gonna half the measure of the arc that it intercepts. So what's 1/2, what is 1/2 times 360 minus 2x? Well, one 1/2 times 360 is 180. 1/2 times 2x is x. So the measure of this angle is gonna be 180 minus x degrees. 180 minus x degrees, and just like that we've proven that these opposite sides for this arbitrary inscribed quadrilateral, that they are supplementary. You add these together, x plus 180 minus x, you're going to get 180 degrees. So they are supplementary.