Short Answer 
The steps to analyze angles in quadrilateral ABCD inscribed in a circle begin by understanding the relationship between arcs and angles, applying the Inscribed Angle Theorem to derive angle measures. Finally, it’s demonstrated that angles A and C, along with angles B and D, are supplementary, confirming the relationships in the quadrilateral.
Step 1: Understand the Relationship Between Arcs and Angles
In quadrilateral ABCD inscribed in a circle, the measure of arc BAD is given as a¬∞. The remaining arc, BCD, can be calculated using the formula for the complete circle, which is 360¬∞. Thus, the measure of arc BCD is 360¬∞ Рa¬∞. This relationship between arcs and the corresponding angles is fundamental to solving the problem.
Step 2: Apply the Inscribed Angle Theorem
The Inscribed Angle Theorem states that an angle inscribed in a circle is half of the corresponding central angle subtending the same arc. For our quadrilateral:
- The measure of angle A is m‚a†A = a¬∞/2.
- The measure of angle C is m‚a†C = (360¬∞ Рa¬∞)/2.
Step 3: Prove the Supplementary Nature of Angles A, C, B, and D
By adding the measures of angles A and C, we find that: m‚a†A + m‚a†C = 180¬∞. This shows that angles A and C are supplementary. Additionally, since the sum of all angles in quadrilateral ABCD is 360¬∞, we set up the equation:
- m‚a†A + m‚a†C + m‚a†B + m‚a†D = 360¬∞
- Substituting gives us 180¬∞ + m‚a†B + m‚a†D = 360¬∞.