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geometry intermediate

Problem

problem

In the adjoining figure

∡ E = 4 0^{\circ}

and arc

A B

, arc

B C

, and arc

C D

all have equal length. Find the measure of

∡ A C D

.

(A)

1 0^{c} i r c l e . s t r o k e d . t in y

(B)

1 5^{c} i r c l e . s t r o k e d . t in y

(C)

2 0^{c} i r c l e . s t r o k e d . t in y

(D)

(45/2)^{c} i r c l e . s t r o k e d . t in y

Solution

If arcs

A B

,

B C

, and

C D

are congruent, then

∡ A C B = ∡ B D C = ∡ C B D = θ

. Because

A B C D

is cyclic,

∡ C A D = ∡ C B D = θ

, and

∡ A D B = ∡ A C B = θ

. Then,

∡ E A D = ∡ E D A = \frac{18 0 ^{\circ} - 4 0 ^{\circ}}{2} = 7 0^{\circ}

.

θ = 5 5^{\circ}

.

∡ A C D = 18 0^{\circ} - 5 5^{\circ} - 11 0^{\circ} = 1 5^{\circ}

.

Final answer

B