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