Quadrilaterals Inscribed in Circles
A quadrilateral is said to be inscribed in a circle if all four vertices of the quadrilateral lie on the circle. Quadrilaterals that can be inscribed in circles are known as cyclic quadrilaterals. The quadrilateral below is a cyclic quadrilateral.
Not all quadrilaterals can be inscribed in circles and so not all quadrilaterals are cyclic quadrilaterals. A quadrilateral is cyclic if and only if its opposite angles are supplementary.
Proving Supplementary Angles
One method of proof is called a proof by contradiction. With a proof by contradiction you prove that something cannot not be true. Therefore, it must be true. Here, you are attempting to prove that it is impossible for a quadrilateral with opposite angles supplementary to not be cyclic. Therefore, such a quadrilateral must be cyclic.
Finding a Point of Intersection
The two highlighted statements are a contradiction -- they cannot both be true. This means that your original assumption cannot exist. You cannot have a quadrilateral with opposite angles supplementary that is not cyclic. So, if opposite angles of a quadrilateral are supplementary then the quadrilateral must be cyclic.
Note: You will learn more about proof by contradiction in future courses!
Solving for Unknown Values
Earlier, you were given a problem about a rhombus.
1. What is a cyclic quadrilateral?
2. A quadrilateral is cyclic if and only if its opposite angles are __________________.
To see the Review answers, open this PDF file and look for section 8.6.