In the adjoining figure ∠ E=40° and arc AB, arc BC and arc CD all have equal length. Find the measure of ∠ ACD.

1977 AMC 12 Problem 9

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In the adjoining figure

\angle E=40^\circ

and arc

AB,

arc

BC

and arc

CD

all have equal length. Find the measure of

\angle ACD.

10^\circ

15^\circ

20^\circ

\left(\frac{45}2\right)^\circ

30^\circ

Assign one variable to each of the three equal arcs and another to arc

AD

Use both the full-circle arc sum and the external-secant angle theorem at

E

Let each of arcs

AB,

BC,

CD

measure

x^\circ

and let arc

AD

measure

y^\circ.

Then

\begin{aligned} 3x+y&=360,\\ \frac{x-y}{2}&=40. \end{aligned}

Solving gives

y=30.

The inscribed angle

\angle ACD

subtends arc

AD,

so it measures

\frac{y}{2}=15^\circ.

Therefore, the correct answer is B.