Proof: Angle between a tangent and a secant
About this video
The video proves that the angle between a secant and a tangent meeting outside a circle is half the difference of the intercepted arcs: x = (1/2)(phi - theta). It draws a chord to form a triangle, uses the tangent-chord angle (theta/2) and an inscribed angle (phi/2), finds the supplementary angle 180 - phi/2, then uses the triangle angle sum.
You'll be able to
- state that the angle between a tangent and a secant meeting outside a circle is half the difference of the intercepted arcs.
- use the tangent-chord angle, inscribed angles and supplementary angles in a proof.
- find the angle or a missing arc using x = (1/2)(far arc - near arc).
Key words
- secant
- A line that crosses a circle at two points.
- tangent
- A line that touches a circle at exactly one point.
- tangent-chord angle
- The angle between a tangent and a chord at the point of tangency; it is half the intercepted arc.
- inscribed angle
- An angle with its vertex on the circle; it is half the intercepted arc.
- supplementary
- Two angles that add up to 180 degrees.
Transcript
- [Instructor] What we're gonna do in this video is prove that if we have a secant and a tangent that intersect outside of the circle, that the angle formed between them, let's say that has a measure of x, is going to be 1/2 the difference of the measures of the arc that they intersect, or the arcs, I should say, that they intersect.
So for example, if this arc has a measure of phi, and this arc has a measure of theta, we wanna prove, and this is a very similar result to the one where we just had two secants intersecting outside of a circle, but we wanna prove that x is going to be equal to 1/2 of the difference.
In this case, it's going to be phi minus theta. Phi minus data. So how do we go about doing that? Well, we've done a similar proof several times. What we could do is construct a little bit of a triangle right over here, like this. And we know that when you have a tangent and a cord like this, the angle that is formed is 1/2 of the measure of the arc that it intersects, right over here.
So this angle right over here is, let me do it like this. This is going to be theta. Theta, that's the measure of the arc that it intersects, theta over 2. Now we also know, we've done this in other videos, that when you have an inscribed angle, and it intersects an arc, the measure of this angle is going to be 1/2 the measure of the arc that it intersects.
So this angle right over here is going to be phi over 2. And what is interesting now is we know that this angle, phi over 2, is supplementary to this angle right over here. I just don't wanna make you think that they're the same angle by just only doing one arc.
So this angle right over here is 180 minus phi over 2. So now we know that this angle plus this angle plus that angle is going to be equal to 180 degrees because they're all in the same triangle. So lemme just write that down. I could write that x plus theta over 2, theta over 2 plus 180 minus phi over 2, 180 minus phi over 2, is equal to 180.
Well, there are several things that I can do. I can subtract 180 from both sides. I can subtract 180 from both sides. I can add phi over 2 to both sides. I'm just trying to isolate the x on the left. So add phi over 2 to both sides, so plus phi over 2, and I can subtract theta over 2 from both sides.
Once again, I'm just trying to isolate the x. Minus theta over 2. And then what do I get? Well that cancels with that. That cancels with that. That cancels with that. We're left with just an x on the left-hand side and on the right hand side, that cancels with that, that cancels with that, and we are left with the result that we were hoping to get.
We get phi over 2, phi over 2 minus theta over 2. If we just factor out 1/2, we get exactly what we set out to prove.
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