Proof that opposite angles of a cyclic quadrilateral are supplementary

A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle. In this type of quadrilateral, opposite angles are supplementary, which means that they add up to 180 degrees. This can be proved in the following steps:

Step 1: Draw a cyclic quadrilateral ABCD

Step 2: Draw a line from the center of the circle to the midpoint of side AB. Let this line be denoted as O1.

Step 3: Draw a line from the center of the circle to the midpoint of side CD. Let this line be denoted as O2.

Step 4: Draw lines from the center of the circle to the vertices of the quadrilateral. Let these lines be denoted as OA, OB, OC, and OD.

Step 5: Observe that triangles O1AB and O2CD are congruent by SAS (side-angle-side) because they have a common side (the line from the center of the circle to the midpoint of AB or CD), their corresponding angles (AO1B and CO2D, and BO1A and DO2C) are equal because they are subtended by the same arc (AB or CD), and their opposite sides (AB and CD) are equal because they are opposite sides of a cyclic quadrilateral.

Step 6: Observe that angles AOB and COD are equal because they are subtended by the same arc (AB) and arcs subtended by the same angle are equal.

Step 7: Observe that angles BO1A and DO2C are also equal because they are corresponding angles of congruent triangles.

Step 8: Together, angles AOB, BO1A, and O1AB form a straight line because they are angles on a straight line.

Step 9: Similarly, angles COD, DO2C, and O2CD form a straight line.

Step 10: Therefore, angles AOB and COD are supplementary because they add up to 180 degrees, and angles BO1A and DO2C are supplementary because they add up to 180 degrees.

Step 11: Hence, opposite angles of a cyclic quadrilateral are supplementary.