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two chords intersect in a circle forming right angles. what is the sum of the measures of a pair of opposite arcs

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If two chords, say AB and then BC intersect at B so that ABC is 90 degrees, then you have a right triangle ABC. Let O be the center of the circle and consider the arc AOC. By the inscribed angle theorem, ½ Angle (AOC) = Angle (ABC). But this means Angle (AOC) is 180 degrees. So A-O-C is a straight line segment. But O is the center, so AC must then be the diameter. So the line segment AC splits the circle into two equal arcs, both of 180 degrees. So Arc AB + Arc BC = 180.
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