Cyclic Quadrilaterals

Cyclic Quadrilaterals

A cyclic quadrilateral has all four vertices on a circle. Its opposite angles are supplementary, so they add to $180^\circ$. The exterior angle equals the interior opposite angle. In proofs, first establish that points are concyclic or that a quadrilateral is cyclic; then use the correct relationship with a named reason.

Worked reasoning

In cyclic quadrilateral ABCD, if $\angle A=112^\circ$, then $\angle C=180-112=68^\circ$ because opposite angles are supplementary.

Exam method

1. Identify opposite versus adjacent angles. 2. Confirm the quadrilateral is cyclic. 3. Use the 180-degree sum or exterior-opposite equality and state the theorem.

Establish cyclicity before using it

Sometimes a question asks you to prove a quadrilateral cyclic. One route is to show a pair of opposite angles is supplementary; another is to show an exterior angle equals the interior opposite angle. Only after that conclusion may you use cyclic-quadrilateral properties in later steps. This forward structure matters in proof questions: a familiar-looking four-sided shape is not automatically cyclic just because it appears inside a circle-like drawing.

One-minute retrieval: Cyclic Quadrilaterals

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.