This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To prove the theorem that states the opposite angles of a cyclic quadrilateral are supplementary, follow these steps: Step 1: Draw a circle with center O and a cyclic quadrilateral ABCD inscribed in it. A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. Step 2: Join the vertices B and D to the center O. This creates two central angles: BOD (subtended by arc BCD) and reflex BOD (subtended by arc BAD). Step 3: Apply the theorem that the angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle. For arc BCD: BOD = 2 BAD (Equation 1) For arc BAD: reflex BOD = 2 BCD (Equation 2) Step 4: Sum the angles around the center O. The sum of angles around a point is 360^: BOD + reflex BOD = 360^ (Equation 3) Step 5: Substitute Equation 1 and Equation 2 into Equation 3. 2 BAD + 2 BCD = 360^ Step 6: Divide the entire equation by 2. BAD + BCD = 180^ This shows that the opposite angles A and C are supplementary. Step 7: Conclude the proof. Similarly, by joining vertices A and C to the center O, it can be proven that ABC + ADC = 180^. Therefore, the opposite angles of a cyclic quadrilateral are supplementary. The proof shows that the sum of opposite angles in a cyclic quadrilateral is 180^. Drop the next question.