Hi @The-Rogue-Blade!

We assumed that there exist four points that have equal angles as shown on picture below, and we want to prove that these four points lie on the one circle by contradiction.

M2D8C4-2.jpg

We know that inscribed angles that subtend the same arc on the circle are equal. So we are drawing an additional line and get one more green angle that equal previous two:

M2D8C4.jpg

So now we have a triangle with interior and exterior angles that are equal. Hence, those two points are the same ones. 🙂

@The-Rogue-Blade Thank you so much for being observant and catching this error!! This has been fixed now. 🙂

@TSS-Graviser The arcs and the angles actually go together. 🙂 An arc is a fragment of the line of a circle. In the picture below, the yellow angle inscribes the yellow arc, because you can see that the ends of the angle's segments meet the ends of the yellow arc.

M2W2-what-about-arcs-and-angles.png

Now, I've drawn another angle that also inscribes the yellow arc. This angle is equal to the first yellow angle!

M2W2-what-about-arcs-and-angles2.png

Isn't that neat? This is also how we are able to prove the Power of a Point Theorem. Do you see that since the yellow angles are the same, we can form similar triangles? It's because ---- look! There are two more angles that inscribe the same arc!

M2W2-what-about-arcs-and-angles3.png

The green angles inscribe the same green arc, and so they are the same as well!

Don't worry if it takes a few times of seeing these ideas before they sink in. It's natural when you are learning something unfamiliar. Remember the saying, "You have to spend 10,000 hours doing anything in order to become an expert." So don't worry if takes time, because this is part of the process of attaining real understanding!

@TSS-Graviser A cyclic quadrilateral is a 4-sided figure that can be drawn inside a circle with its corners touching the circle, like this:

M2W2-cyclic-quad.png

You'll see in the lesson that opposite angles of the cyclic quadrilateral add up to \( 180^{\circ}\). There is also something else called Ptolemy's Theorem, which we don't learn in Module 2, but later on in Module 8 (which isn't available yet). This says that the sum of the products of pairs of sides equals the product of the diagonals. Cyclic quadrilaterals are coooool!