Cyclic Quadrilaterals and Parallel Lines in Circles - Concept

A cyclic quadrilateral is an inscribed quadrilateral where the vertices are all on the circle and there exists a special relationship between opposite angles in the cyclic quadrilateral, so let's start off by looking at angle b and angle d. Well I know that the measure of angle d in terms of the intercepted arc is that it's always going to be half, that's the definition of an inscribed angle, so we have half arc abc.
We also know that its opposite angle b, since angle b is an inscribed angle that's going to be half of its intercepted arc which is adc. And if I look at arc adc and arc abc they form a full circle, so I can write over here that arc abc plus arc adc must sum to 360 degrees. But that doesn't tell us anything specifically about opposite angles, so let's see what happens if I add it up angle d angle b so I'm going to say measure of angle d plus measure of angle b is equal to, I'm just going to substitute in, we have one half arc abc plus one half arc adc. Well if I solve this equation from one of these variables then I can substitute in so using a different color I'm going to subtract arc abc from both sides so now I've solved for arc adc and it's 360 minus arc abc so what I'm going to do is instead of writing adc I'm going to write 360 minus arc abc so this is equal to one half and actually what I'm going to do is I'm going to erase adc and I parenthesis 360 minus arc abc so we still have d plus b is equal to one half arc abc and distributing this one half, half of 360 is 180 minus because we have a positive one half times a negative, minus one half arc abc so if I combine like terms positive one half arc abc negative one half arc abc those are going to add up to 0 so I see I'm going to move over here that measure of angle d plus measure of angle b is equal to 180 degrees so in a cyclic quadrilateral, your opposite angles will always be supplementary.
Another key facet of circles is if you have parallel lines so what I'm going to do is I'm going to draw in a transversal. Something special is going to happen, if I call this x degrees and if I call this y degrees I see that if these two are parallel then I have created alternate interior angles that are congruent and I see that I have an inscribed angle whose vertex is right here and whose intercepted arc is y and over here I have an inscribed angle whose vertex is right here and whose intercepted arc is x. Now if these 2 inscribed angles are congruent then x must equal y, so when you have parallel lines intercepting a circle it will create two congruent arcs so using these 2 facets of circles we can solve for missing angles.