In this problem, we need to find the value of angle x. We're given that P is the center of the circle, and there are two 20-degree angles marked at points B and C. Let's solve this step by step.
First, since P is the center of the circle, the lines PA, PB, and PC are all radii of the circle. This means they all have equal length.
In triangle APB, since PA equals PB, it's an isosceles triangle. This means the angles at A and B are equal, both 20 degrees. Similarly, in triangle BPC, the angles at B and C are both 20 degrees.
Looking at angle ABC, we can see it's made up of two parts: angle ABP plus angle PBC. Since both of these angles are 20 degrees, the total angle at B is 40 degrees.
Now, we can use an important property of circles: the central angle is twice the inscribed angle that subtends the same arc. The angle ABC is an inscribed angle of 40 degrees, and it subtends the same arc as the central angle x.
Therefore, the central angle x equals 2 times 40 degrees, which is 80 degrees. So the answer is 80 degrees.
Let's explore the Inscribed Angle Theorem, which is key to solving our problem. This theorem states that an angle inscribed in a circle is half the central angle that subtends the same arc.
In our diagram, angle ABC is the inscribed angle, and angle APC is the central angle. Both angles subtend the same arc AC. According to the theorem, the inscribed angle is exactly half the central angle.
Let's see how this relationship works as we change the central angle. Notice that the inscribed angle is always half the central angle, regardless of where point B is located on the circle.
Applying this theorem to our original problem, we found that the inscribed angle at B was 40 degrees. Therefore, the central angle x must be 2 times 40 degrees, which equals 80 degrees.