Points A, B, C divide the circle centered on O into three arcs: ˘AB, ˘BC and ˘AC, whose degree measures are 7: 5: 6. Find the angles ABC, BAC, AOB.
Let the degree measure of the arc AB = 7 * X0, then the degree measure of the arc BC = 5 * X0, and the arc AC = 6 * X0.
The full circle is 360 degrees.
Then AB + BC + AC = 5 * X + 7 * X + 6 * X = 180.
18 * X = 360.
X = 360/18 = 20.
Then arc AB = 7 * 20 = 140, arc BC = 5 * 20 = 100, arc AC = 6 * 20 = 120.
The inner angles of the triangle ABC are inscribed angles in a circle.
Then the angle BAC = BC / 2 = 100/2 = 50, angle, angle ABC = AC / 2 = 120/2 = 60.
The central angle AOB rests on the arc AB, then the angle AOB = 140.
Answer: Angle ABC = 60, angle BAC = 50, angle AOB = 140.
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