Short Answer 
The discussion explains that a cyclic quadrilateral, such as ABCD inscribed in circle O, follows the Inscribed Angle Theorem, where an inscribed angle is half the measure of its corresponding arc. Additionally, it establishes an equation relating the angles and arcs of the quadrilateral, confirming the cyclic nature of ABCD.
Step 1: Understand Cyclic Quadrilateral ABCD
A cyclic quadrilateral is a special type of four-sided figure where all vertices touch the circumference of a circle. In this case, quadrilateral ABCD is inscribed in circle O. Understanding this concept is crucial because it sets the foundation for applying the Inscribed Angle Theorem later in the analysis.
Step 2: Apply the Inscribed Angle Theorem
The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of the intercepted arc. Here, angle mBCD can be expressed as twice the angle mA:
- mBCD = 2(mA)
- This means that angle mA is related to the arc mBCD through: mA = 1/2 mBCD.
Step 3: Relate the Arcs to Form an Equation
The sum of all arcs in a circle equals 360 degrees. For the arcs of quadrilateral ABCD, we formulate the equation:
- mBCD + mDAB = 360°.
- Using our findings from Step 2, substitute to get 2(mA) + 2(mC) = 360°.
- This shows how angles relate to the arcs, confirming the cyclic nature of quadrilateral ABCD.