Inscribed Angles in Circles Lesson Geometry Concepts

In this video, we are going to learn about inscribed angles in circles. First, let's review what central angle is. If you have circle, central angle is the angle formed by the center of the circle and two points on the circle. So, this would be one example of central angle. And remember that the measure of the ark that is intercepted. So this arc right here is always the same measure as the central angle here. Now if instead of connecting these two points on the circle to the center, we connected them to point across the circle like this. This angle created right here is called an inscribed angle. And an inscribed angle is always half the measure of the ark. The central angle is the same as the ark. The inscribed angle is half the measure of the ark. Okay? So for example, if this arc was 100°, then this angle, the central angle would also be 100°, but the inscribed angle would be 50°. Now, the cool thing about inscribed angles is if you have circle and you pick two points on the circle to make your inscribed angle from, there are an infinite number of inscribed angles that you can draw from those two points. For example, you could have an angle over here. That's one inscribed angle. Or you could have an angle down here. Or you could have an angle over here. And as you can see, you could keep going and going and create more and more angles. And because all of those angles intercept the same arc up here, it means that they all must be congruent because they're all half the measure of that same ark. So that means if this arc was 90°, then each of these angles would have to be 45°. Half of that arc. Also, if you didn't even know what this ark was to begin with, you would know that all of these angles have to be the same. Now, what if you had circle and the arc that you were thinking about was actually semicircle. It goes through the center of the circle. So, you're thinking about this arc and an inscribed angle from those two points. So for example, this right here, because semicircle is 180°, it means that this inscribed angle has to be right angle. So in general, inscribed angles of semicircles are always right angles. And this is important to know because you'll oftent times see picture of circle and you'll know where the diameter is. And you'll see an angle that's been created in half the circle. And it won't tell you that this is right angle. But you have to realize that it has to be because it's an inscribed angle for half the circle for semicircle. So it has to be 90° which is half of 180°. So in general what you have to remember is inscribed angles are always half the measure of their arcs. And that means that any inscribed angle from the same two points will always be congruent to any other inscribed angle from those two points. And if the arc happens to be semicircle then that inscribed angle will always be right angle.