7.4 Cyclic Quadrilaterals

I) What is a Cyclic Quadrilateral Quadrilateral: a polygon with four sides Square, rectangle, parallelogram, trapezoid….. The sum of all the interior angles is equal to 360 degrees Cyclic Quadrilateral (CQ) A quadrilateral with all four vertices (corners) on the circumference of the circle

EX: Find & Name all the CQ’s

II) Properties of a CQ Opposite interior angles in a CQ add to 180 degrees Then opposite angles must be “Supplementary” If ABCD is a CQ If opposite angles are “Supplementary” Then ABCD must be a CQ

III) Proving Opposite Angles in a CQ Add to 180 Prove:

Exterior angles: angles created by the extension of one side The Exterior angle is equal to the opposite interior angle Exterior Angle

Practice: Given: ABCD is a CQ Prove: Statement Reason

EX: Determine the value of each angle

What is the value of “x+y”?

IV) Quadrilaterals & CQ’s Four sides Sum of all interior angles = 360 degrees Not all quadrilaterals are CQ’s Cyclic Quadrilaterals (CQ’s) All 4 vertices are on the circumference Opposite angles are suppl. A quadrilateral can only be a CQ if opp. angles add to 180 degrees NOTE: To prove that a quadrilateral is a CQ, then prove a pair of interior angles to be supplementary

Practice: Prove OACD is a CQ Statement Reason

Statement Reason Prove: Given: G is the midpoint of AB E is the midpoint of AC Prove: Statement Reason

Ex: Statement Reason

Given that ∆ABC is a right triangle and EDB is 90o Given that ∆ABC is a right triangle and EDB is 90o. Prove that Angle DBE and ECD are equal.