Master Circle Theorems in Minutes! 💡 | Find Any Unknown Angl e Fast

Hi guys, welcome to today's video. Today we'll be looking at an introduction to circuit theory and some of the rules that might apply. We have this beautiful shape with different angles that we need to find and different points on the circumference. And so we're going to see if we can find all of them easily and quickly. The first unknown that we're going to try to find is x. The rule that applies in this case is that the angle at the center is twice the angle at the circumference. That's if they're facing the same side. What do I mean? If you look at the curve here and you look at the curve here they're both facing the same side right? So the angle at the center which is 105 degrees is twice the angle at the circumference which is our x. So to find x all we have to say is x is half of 105 degrees. x would then be 52.5 degrees. The next unknown we're going to try to find is w. For w the same rule applies but in this case we're going to take the other side because that's the side that's facing the same direction as w. We're not going to use that rule though. We're going to use a different rule. The rule is that for the opposite sides of a quadrilateral in a circle they add up to 180 degrees. What does this mean? x and w are opposite each other therefore x plus w equals 180 degrees. Since we already have our x we'll say w is 180 minus 52.5. That brings our w to 127.5 degrees. Now the third unknown that we have is y. Even though y is outside the circle it still has a lot of relationship with some elements in the circle. To find our y we're going to apply two theorems. The first of them is that angles in the same segment are equal. How do we apply this? If you notice the angle at a and the angle at e are both in the same segment because they're going in the same direction and they stem from the same points b and d. Therefore this angle is also 52.5 degrees. The second rule we're going to use is that in a triangle the value of the exterior angle is equal to the sum of the two opposite interior angles. This may seem a little confusing at first but I'm going to show it to you in a different way. For any triangle that you have this angle a equals the sum of b and c. If you notice the angle at e in the circle 52.5 is the exterior angle and that will mean that it is the sum of y and this other angle here. Our work has been made easy because we know that the angle at f which is y is equal to the other angle at d. How do we know this? Because this mark here shows us that it's an isosceles triangle. You wonder how this might help us? Well it simply means that this angle is also y and that 52.5 equals y plus y. y would then be equal to 52.5 divided by 2. y would be 26.25 degrees. Great job finding all the unknowns. I hope you enjoyed it as much as I did. If you enjoyed this video or found it helpful please like it and subscribe to the channel for more math content.