1. ## Geometry proof dealing with quadrilaterals inscribed in a circle

Hey everyone, I'm a little stuck on this problem -- any help would be very much appreciated.

i) Prove that if a quadrilateral is inscribed in a circle, then the sum of its oppsoite angles is 180 degrees.
(in other words, prove: m<A + m<C = m<B + m<D = 180 degrees)

and part two of the question is -- What kinds of parallelograms can be inscribed in circles?

I'm thinking that it is ALL parallelograms but it would be nice to verify that. Thanks for any help

2. Actually it is true the other way around!
(Without lose of generality).
Anyway.

3) 2*<ADC + 2*<ABC = arc ABC + arc ADC = 360

3. thanks for the help . As for the 2nd part, is this only true for quadrilaterals (shapes adding up to 360 degrees) or would it be all paralellograms?

4. Originally Posted by WzMath16
thanks for the help . As for the 2nd part, is this only true for quadrilaterals (shapes adding up to 360 degrees) or would it be all paralellograms?
No.

This excerise shows the necessary conditions. Meaning if a quadrilateral is cyclic (meaning can be insribed in circle) then its opposite angles MUST add up to 180. Thus, the only possibilities are the ones that opposite angles add up to 180.

(But as I said it also happens to be sufficient. Meaning if a quadrilateral opposite angles add to 180 then it is cyclic. But that is much more complicated to show. Thus, together we have necessary and sufficient condtitions, that is those and only thus with opposite angles that are supplemantry).

Originally Posted by WzMath16
thanks for the help . As for the 2nd part, is this only true for quadrilaterals (shapes adding up to 360 degrees) or would it be all paralellograms?

You should be able to figure it out.

Move the red points to make any kind of cyclic quadrilateral.