Problem: Points A , B , Q , D A, B, Q, DA , B , Q , D , and C CC lie on the circle shown and the measures of arcs B Q ^ \widehat{B Q}B Q β and Q D ^ \widehat{Q D}Q D β are 4 2 β 42^{\circ}4 2 β and 3 8 β 38^{\circ}3 8 β respectively. The sum of the measures of angles P PP and Q QQ is
Answer Choices:
A. 8 0 β 80^{\circ}8 0 β
B. 6 2 β 62^{\circ}6 2 β
C. 4 0 β 40^{\circ}4 0 β
D. 4 6 β 46^{\circ}4 6 β
E. None of these
Solution:
β P + β Q = 36 0 β β ( β P A Q + β P C Q ) \angle P+ \angle Q=360^{\circ}-(\angle P A Q+ \angle P C Q)β P + β Q = 3 6 0 β β ( β P A Q + β P C Q )
= 36 0 β β ( 18 0 β β 2 1 β ) β ( 18 0 β β 1 9 β ) = 4 0 β . =360^{\circ}-\left(180^{\circ}-21^{\circ}\right)-\left(180^{\circ}-19^{\circ}\right)=40^{\circ} .= 3 6 0 β β ( 1 8 0 β β 2 1 β ) β ( 1 8 0 β β 1 9 β ) = 4 0 β .