Now we can talk about some circle properties and many of the properties that we will talk about in this video will involve angle in circles. The first one that we are gonna talk about is about lengths and this is relatively easy. Any triangle with two sides that are radii have to be isosceles. Now that is pretty obvious when you think about it. Read full transcript
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 sides are equal the angles have to be equal. And so that we have an angle 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 are told that that cord EF, also has a length of R in addition to a few radii, well then immediately we know we have a equilateral triangle and we have 360 degrees angles. Alternately, if we are given two radii and a 60 degree angle between them.
We know if we draw the third cord 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 intersections. One way to talk about the size of an arc is to talk about it's 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 is the angle all the way around a circle. And so just to think about it, if you are standing one way and turn all the way around so that when you stop you are facing the same way again that is what it is to turn 360 degrees.
It is often good to understand these angles in terms of how often you would 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 were told that the arcs are equal we could deduce that the angles were 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 it's 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 chords AB and BC, two chords 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 inscribed 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 that rule is true. Let's look at this diagram now of course in this diagram the gold angle BOC is the central angle.
And the red angle at A, BAC that is the inscribed angle and we'd like to figure out the relationship for 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 two radii sides.
So those two red angles have to be equal and remember that one of them is the inscribed angle. So, the measure of that red angle is a 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 arch? 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 can add up the three angles in the triangle, 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 2x = BOC and from here we can just divide by 2. So x = one-half the central angle, and of course the central angle equals the arc. So it equals one-half 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 intercepts a semicircle, that is, any inscribed angle that intercepts the endpoints of a diameter has to be a right angle. Because HG is a diameter, that's a diameter, so it means that this arc here is a semicircle, a 180 degree arc.
Well, the angle HKJ intersects that arc, so it has to be half the measure of that arc. Well 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, and 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 angel, that would be a very different angle.
In fact, as it happens that would be supplementary to 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 cord, 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 that 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 intercept 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 intercepts 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, it 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 lookout for equilateral built of radii inside a circle. A central angle which is an angle which has the vertex of the center has the same measures as the arc it intercepts. Equal chords intersect equal arcs.
An inscribed angle has half the measure of the arc it intercepts. Remember, an inscribe 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 the 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.