Conjectures in Geometry: Inscribed Angles
Time-saving video on how to define and label central angles and their intercepted arcs. Example problems find missing central angles and missing arc . It is the central angle's ability to sweep through an arc of degrees that determines the number of degrees usually thought of as being contained by a circle. Central angles are angles formed by any two radii in a circle. Degree measure of a minor arc: Defined as the same as the. A circle's central angles and the arcs that they cut out are part of many circle proofs. They also come up in many area problems. The following figure shows how.
And let's say that this is the other ray. This right over here is the other ray of the angle. I encourage you to pause this video and try to figure out what the measure of this angle right over here is. Well, let's think about where the rays intersect the circle. They intersect there and there.
Inscribed Angles Conjectures
The arc that connects them on the circle is that arc right over there. That is literally half of the circumference of the circle. That is half of the circumference, half of the way around of the circle, circumference of the circle. So this angle is going to be half of degrees. And half of is degrees. And when you view it this way, these two rays share a common endpoint. And together, they're really forming a line here. And let's just do one more example, because I said I would.
Let me paste another circle. Let me draw another angle. So let's say that's one ray of the angle, and this is the other ray. This is the other ray of the angle right over here. There's actually two angles that are formed. There's actually two angles formed in all of these. There's one angle that's formed right over here, and you might recognize that to be a degree angle.
But what we really care about in this example is this angle right over here. So once again, where does it intersect the circle?
We care about this arc right over here, because that's the arc that corresponds to this angle right over here. So we get psi -- this psi plus that psi plus psi plus this angle, which is minus theta plus minus theta. These three angles must add up to degrees. They're the three angles of a triangle. Now we could subtract from both sides. Si plus psi is 2 psi minus theta is equal to 0.
Arc Length and Radian Measure - MathBitsNotebook(Geo - CCSS Math)
Add theta to both sides. You get 2 psi is equal to theta. So we just proved what we set out to prove for the special case where our inscribed angle is defined, where one of the rays, if you want to view these lines as rays, where one of the rays that defines this inscribed angle is along the diameter. The diameter forms part of that ray.
So this is a special case where one edge is sitting on the diameter. So already we could generalize this. So now that we know that if this is 50 that this is going to be degrees and likewise, right? And now this will apply for any time.
We could use this notion any time that -- so just using that result we just got, we can now generalize it a little bit, although this won't apply to all inscribed angles.
Let's have an inscribed angle that looks like this. So this situation, the center, you can kind of view it as it's inside of the angle. That's my inscribed angle.
And I want to find a relationship between this inscribed angle and the central angle that's subtending to same arc. So that's my central angle subtending the same arc. Well, you might say, hey, gee, none of these ends or these chords that define this angle, neither of these are diameters, but what we can do is we can draw a diameter.
If the center is within these two chords we can draw a diameter. We can draw a diameter just like that. If we draw a diameter just like that, if we define this angle as psi 1, that angle as psi 2. Clearly psi is the sum of those two angles. And we call this angle theta 1, and this angle theta 2. So si, which is psi 1 plus psi 2, so psi 1 plus psi 2 is going to be equal to these two things.
Psi 1 plus psi 2, this is equal to the first inscribed angle that we want to deal with, just regular si. What's theta 1 plus theta 2? Well that's just our original theta that we were dealing with. So now we've proved it for a slightly more general case where our center is inside of the two rays that define that angle.
Now, we still haven't addressed a slightly harder situation or a more general situation where if this is the center of our circle and I have an inscribed angle where the center isn't sitting inside of the two chords.
Let me draw that. So that's going to be my vertex, and I'll switch colors, so let's say that is one of the chords that defines the angle, just like that. And let's say that is the other chord that defines the angle just like that. So how do we find the relationship between, let's call, this angle right here, let's call it psi 1. How do we find the relationship between psi 1 and the central angle that subtends this same arc? So when I talk about the same arc, that's that right there. So the central angle that subtends the same arc will look like this.
Let's call that theta 1. What we can do is use what we just learned when one side of our inscribed angle is a diameter. The minor arc is the smaller of the two arcs, while the major arc is the bigger.
We define the arc angle to be the measure of the central angle which intercepts it. The Inscribed Angle Conjecture I gives the relationship between the measures of an inscribed angle and the intercepted arc angle. It says that the measure of the intercepted arc is twice that of the inscribed angle. The precise statements of the conjectures are given below. Each conjecture has a linked Sketch Pad demonstration to illustrate its truth proof by Geometer's Sketch Pad!
The linked activities sheet also include directions for further "hands on" investigations involving these conjectures, as well as geometric problems which utilize their results. The precise statement of the conjectures: