4
$\begingroup$

There is a well-known theorem that a cyclic quadrilateral (its vertices all lie on the same circle) has supplementary opposite angles.

I have a feeling the converse is true, but I don't know how to prove it. The converse states:

If a quadrilateral's opposite angles are supplementary then it is cyclic.

Should I approach this proof by contradiction? Or is it possible to prove by construction?

  • 0
    Show that one vertex lies on the circle determined by the other three. If you don't already "know" that a converse of the Inscribed Angle Theorem (rather, one of its immediate corollaries) gives this to you, then, sure, you could use contradiction effectively to prove the needed converse. But it's also possible to find a related direct approach.2012-02-29
  • 0
    Can anyone prove this by construction? That is, construct a quadrilateral whose opposite angles are supplementary, then prove it is cyclic?2012-09-02

2 Answers 2