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Now we can talk about some circle properties. And many of the properties that we’ll talk about in this video involve angles in circles. The first one we’re gonna talk about actually is about lengths, and this is relatively easy. Any triangle with two sides that are radii has to be isosceles.
Now that’s probably pretty obvious when you think about it. Of course, if we look at this triangle here, OB and OA are radii, so of course they’re equal, because all radii of the same circle are equal. Well, right away that means we have a triangle with two equal sides, in other words, an isosceles triangle. And because the angles are, the sides are equal, the angles have to be equal.
And so that we have an of 70 degrees at A so that means we have to have an angle of 70 degrees at B, which leaves 40 degrees for the angle at O. If the chord side of such a triangle is also equal to the radius then the triangle would be equilateral, which of course is a special case of isosceles. So if we’re told that that chord, EF, also has a length of r in addition to the three radii, well, then immediately we know we have an equilateral triangle and we have three 60 degree angle.
Alternately, if we’re given two radii and a 60 degree angle between them, we know if we draw the third chord we will have an isosceles triangle. So that necessarily means. That that third chord would have to have a length equal to the length of the radius. Now, notice that that angle at the middle, angle EOF, has its vertex at the center of the circle.
This is a very special kind of angle. An angle with its vertex at the center of the circle is called a central angle. A central angle has a unique relationship with the arc it intersects. One way to talk about the size of an arc is to talk about its arc measure, that is, how many degrees it has. The measure of a central angle equals the measure of the arc.
So, for example, here we have this arc. That goes from J to K to L and we have the angle because the angle equals 135 degrees. That automatically means that the arc from J to K to L is also a 135 degree arc. Now, a diameter is essentially a 180 degree angle. As we go from A to O to B, it’s a straight line.
We don’t bend at all. So, that’s 180 degrees on each side. And so that divides the circle into two 180 degree arcs. And, of course, an arc with a measure of 180 degrees, we call a semicircle. The measure of the entire circle, all the way around the circumference, is 360 degrees, that’s the angle all the way around a circle.
And so just to think about it, if you’re standing one way and turn all the way around so that when you stop you’re facing the same way again, that’s what it is to turn 360 degrees. It’s often good to understand these angles in terms of how much you would actually have to turn yourself. If two different central angles in the same circle have the same measure they will intersect arcs of the same size.
So we have two equal angles there, so they intersect two equal arcs. And if we’re told that the arcs are equal, we can deduce that the angles are equal. Similarly, equal length chords intersect equal length arcs. So if we know that the chords have equal length then the arcs have to be equal. Again, if we were told that the arcs were equal we could deduce that the chords were equal.
So central angles have their vertices at the center of the circle. Another kind of angle has its vertex on the circle, point B is on the circle. This kind of angle is called an inscribed angle. The sides of an inscribed angle are always two chords that meet at the vertex. So, this angle is formed by the cords, AB, and BC. Two cords, and they share a common endpoint B, that common endpoint is the vertex of the inscribed angle.
An inscribed angle also has a special relationship with the arc it intercepts, a different relationship. The measure of the inscribed angle is half the measure of the arc it intercepts. So for example, here we have an ascribed angle of 40 degrees that has to be half the arc, so the arc has to be 80 degrees. Here’s one way to see why the rule is true.
Let’s look at this diagram. Now of course, in this diagram the gold angle, BOC, is the central angle. And one of the, the red angle at A, BAC, that is the inscribed angle and we would like to figure out the relationship of that inscribed angle, that red angle, to the arc. Well, certainly the central angle equals the arc.
That’s easy. Okay? So the arc equals that gold central angle. We know that that triangle, ABO, has to be an isosceles triangle because it has to radii sides, so those to red angles have to be equal. And remember that one of them is the inscribed angle, so the measure of that red angle is the measure of the inscribed angle.
We’re just gonna call that x, that’s the thing we’re looking for. How does that x relate to the size of the arc? Well, certainly it’s true, if we add the gold and the blue, those two angles that lie along a straight line, of course they have to add up to 180. Because any two angles on a straight line have to add up to 180. Also, we could add up the three angles in the triangles.
So, that would be 2x, the two red angles, plus the blue angle, that also has to equal 180. The sum of any three angles in a triangle have to equal 180. Well take a look at this, we have thing plus AOB equals 180. Thing plus AOB equals 180. Well, right away if we have gold plus blue equals 180 and red plus blue equals 180, well, that must mean that red and gold equal the same thing.
So in other words, what we have is 2x equals BOC. And from here, we can just divide by two, so x equals 1/2 the central angle. And of course, the central angle equals the arc, so it equals 1/2 the arc. The measure of the inscribed angle equals half the arc. So that is one way to see this is true.
If you remember this argument, it really will help you remember the fact much more deeply. This means that any inscribed angle that intersects a semicircle, that is any inscribed angle that intersects the endpoints of a diameter, has to be a right angle.
Because HJ is the diameter. That’s a diameter so it means that this arch here, is a semicircle, a 180 degree arch. Well, the angel, HKJ, intersects that arch so it has to be have the measure of that arch. Well, half, half of 180 degrees is 90 degrees and that must mean that at K, we have a 90 degree angle.
The test absolutely loves this particular fact. They love it because once you have a right triangle then you can use the Pythagorean theorem. You can use all kinds of ratios, all kinds of things you can do once you know you have a right angle. If two inscribed angles in the same circle intercept the same arc or the same chord and intersect the same chord on the same side, then the two inscribed angles are equal.
So, here we have two inscribed angles. Intersecting that chord LM, then these two must be equal. Now they have to intersect on the same side of the chord. If we also drew an angle over here, on the other side, obviously that would be a much wider angle, that would be a very different angle. In fact, as it happens, that would be supplementary to the, the two angles at M and P, but we don’t need to worry about that.
All you need to know is, as long as the angles are on the same side of the chord, then the two inscribed angles have to be equal. Finally, we will discuss a line, a very special kind of line, outside the circle. A tangent line is a line that passes by a circle and just touches it at only one point.
The word tangent actually means to touch. We get the English word tangible from this same root. If we draw a radius to the point of tangency, then the radius and the tangent line are perpendicular. So this is another case where the geometry itself is guaranteeing that we have a right angle.
Here for clarity, I drew the little perpendicular sign, but even if that sign were not drawn, as long as you know you have a radius to the point of tangency of a tangent line, you know you have a right angle there. And of course, this also lends itself to all the other special right-angle facts, the Pythagorean theorem, the ratios, all that stuff. Here’s a practice problem.
Pause the video and then we’ll talk about this. Okay. So we have a relatively complex diagram here. First of all notice that angle PQS and PRS, these two angles intersect the same chord, PS.
So two angles that intersect the same chord in the same circle, have to be equal. That means that PRS. Has to have a measure of 40 degrees as well. Well, now notice that triangle PSR is a right angle, because PR is a diameter, so PSR is a right angle.
Well, we have 40 degrees at R, we have 90 degrees at S, means we have to have 50 degrees at P. So that angle SPR is a 50 degree angle. Now notice, look at the tangent line. The tangent line intersects a radius and P is the point of tangency. So we have a right angle, TPO, is a right angle.
And it’s comprised of these two smaller angles. One is the 50 degree angle, SPR. And so that’s the 50 degree angle, means that the leftover angle, TPS, the angle we’re looking for, that has to be a 40 degree angle. And so, TPS equals 40 degrees. In summary, if two sides of a triangle are radii, then the triangle is isosceles and of course always be on the look out for equilateral triangle built of radii inside a circle.
A central angle, which is an angle which has its vertex at the center, has the same measure as the arc it intercepts. Equal chords intersect equal arcs. An inscribed angle has half the measure of the arc it intercepts. Remember an inscribed angle has its vertex on the circle. An angle inscribed in a semicircle is 90 degrees.
That’s a very handy fact to keep in mind. Two inscribed angles intersecting the same chord on the same side are equal. And, a tangent line is perpendicular to the radius at the point of tangency. So, notice there are two cases here, the angle inscribed in a semicircle, and the angle between a tangent line and a radius. Both cases where the geometry itself guarantees that we have a right angle, and therefore opens up the Pythagorean Theorem and all the other special right angle facts.
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