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Points A, B, C, and D lie on circle M. Line segment BD is a diameter.
Triangle A C D is inscribed within circle M. Point B is on the circle between points C and B. A line is drawn to connect points D and B. Lines are drawn from points C and A to point M to form a right triangle. Arcs C D and A D are congruent.
What is the measure of angle ACD?
45.0°
67.5°
112.5°
135.0°
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67.5°
Reason: Let arc AD = arc CD = x. Then arc AC = 360° − 2x. Because AM and CM are radii, ∠AMC is the central angle for arc AC; it is given to be a right angle, so arc AC = 90°. Thus 360° − 2x = 90°, so x = 135°. The inscribed angle ∠ACD subtends arc AD, so ∠ACD = 1/2 · arc AD = 135°/2 = 67.5°.
Reason: Let arc AD = arc CD = x. Then arc AC = 360° − 2x. Because AM and CM are radii, ∠AMC is the central angle for arc AC; it is given to be a right angle, so arc AC = 90°. Thus 360° − 2x = 90°, so x = 135°. The inscribed angle ∠ACD subtends arc AD, so ∠ACD = 1/2 · arc AD = 135°/2 = 67.5°.
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