Proof: Angle formed by intersecting chords
At about 103s the narrator names the inscribed angle 'ADC' and then 'ADB'; minor label slip, not tested.
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The video proves that the angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs: x = (theta + phi)/2. It draws chord AD to make a triangle, uses the inscribed angle theorem to get angles of theta/2 and phi/2, finds the third angle from the triangle sum, and uses the fact that this angle and x are supplementary.
You'll be able to
- state that the angle between two intersecting chords is half the sum of the intercepted arcs.
- use the inscribed angle theorem and the triangle angle sum inside a proof.
- find the angle between intersecting chords given the two arc measures.
Key words
- chord
- A segment joining two points on a circle (spelled "cord" in the captions).
- intercepted arc
- The arc that lies inside an angle, between the points where the angle's sides meet the circle.
- inscribed angle
- An angle with its vertex on the circle; its measure is half its intercepted arc.
- supplementary angles
- Two angles that add up to 180 degrees, such as angles on a straight line.
- vertical angles
- Opposite angles formed by two crossing lines; they are equal.
Transcript
- [Narrator] So what we wanna prove in this video is that the angle formed by intersecting cords. So we have two intersecting cords right over here. And so the angles formed by those intersecting cords, we could pick different angles, but let's say we looked at this angle here that has a measure of X.
And of course, that would be the same as this angle here. Because they're vertical angles, that also has a measure of X. Let's just focus on this one up here. What we want to do is prove that X, the measure of that angle, is equal to half the sum of the measures of the intercepted arcs.
So what do I mean half the sum of the measures of the intercepted arcs? So let's say the measure of this arc is theta and the measure of this arc right over here is phi. So then we wanna prove that the measure of this angle, this angle, formed by the intersecting cords, AE and BD.
So that measure right there, that it's equal to half the sum of the measure of the intercepted arcs. So theta, lemme do that in the red color, theta plus phi. So if you are so inspired, I encourage you to pause the video and have a go at it. Otherwise I will proceed.
Now to help us do this, I will set up a new triangle. I'll draw another cord that goes from A to D, just like that. And in other places, we have shown that if you have an inscribed angle, so for example angle ADC. So in other places, we have shown that the measure of an inscribed angle, in this case ADB, is going to be half the measure of the arc that it intersects.
So it's going to be half of theta. So the measure of this, let me just write this over here. Just to make it clear, it's not the same as that one. I'll do two over here. This is one half of theta. And by that same logic, this angle over here is going to be one half of phi, one half of phi.
I could write over here, measure of angle DAC, DAC is equal to one half of phi. Now what's going to be the measure of, I'm looking for a new color here. What's going to be the measure of this angle right over here? Well we know that the sum of the angles of a triangle add up to 180.
So it's going to be 180 minus one half phi, or one half theta, I should say, minus one half theta, minus one half phi, minus one half phi. Now what's X going to be? Well, X is going to be, notice X and this angle over here, I guess I'll call it this orange angle, X and this orange angle are supplementary.
They add up to 180 degrees. So another way I could say this, if I took this angle, this orange angle, and if I added X to it, that should be equal to 180 degrees. Well now we just do a little bit of algebra. We can subtract 180 degrees from both sides and we could get, and I'll just do this in one color.
Now we have negative one half theta minus one half, that gonna be a phi over there. I'm getting my thetas in, minus one half phi, minus one half phi, plus X is equal to zero. And then what we could do is we could add, well we could do it either way. We could add these to both sides or add, lemme just do it this way.
So plus one half theta plus one half phi. I'm gonna do add that over here plus one half theta plus one half phi. Whoop, you can't see that, so let me do it a little bit. So plus one half theta plus one half phi. And so what are we left with? We're left with X is equal to one half theta plus one half phi, which is the same thing as one half theta plus phi.
And we are done.
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