Question

A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles expressed in degrees is:

• A. $1080$
• B. $900$
• C. $720$
• D. $540$

Answer 1

Answer: (D)

$540$

If angle at center, i.e., $\angle AOB=20°$

Then angle $\angle AMB=\frac{1}{2}\angle AOB=0°$

AA'BM is a cyclic quadrilateral

$\therefore \angle A{A}^{\prime }B=180°-\angle AMB=180°-\theta$

$\therefore$ sum of angles made by sides of quadrilateral at center = $360°$

Sum of the angles inscribed in four segments = $\sum 180°-\theta =4\left(180°\right)-\sum \theta =720°-180°=540°$