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A cyclic quadrilateral has vertices on the same circle and is inscribed in the circle. The opposite angles have the same endpoints (the other vertices) and together their intercepted arcs include the entire circle. Since the measure of an inscribed angle is half the intercepted arc, the sum of the opposite angles must be 180 degrees.

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.

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.

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