Points A, B, Q, D, and C lie on the circle shown, and the measures of arcs BQ and QD are 42° and 38°, respectively. The sum of the measures of angles P and Q is:

1971 AMC 12 Problem 5

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Points

A,

B,

Q,

D,

and

C

lie on the circle shown, and the measures of arcs

BQ

and

QD

are

42^\circ

and

38^\circ,

respectively. The sum of the measures of angles

P

and

Q

is:

80^\circ

62^\circ

40^\circ

46^\circ

None of these

Use the external-secant formula for angle

P

Adding angle

Q

cancels the unknown intercepted arc

Let

m\overset{\frown}{AC}=u.

The external-secant formula gives

m\angle P=\frac12\left(m\overset{\frown}{BD}-u\right),

while the inscribed-angle theorem gives

m\angle Q=\frac{u}{2}.

Therefore

\begin{aligned} m\angle P+m\angle Q &=\frac12m\overset{\frown}{BD}\\ &=\frac12(42^\circ+38^\circ)\\ &=40^\circ. \end{aligned}

Therefore, the correct answer is C.