Re: Quadrilateral ABCD is inscribed in a circle. What is x - y?
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30 Jan 2023, 11:21
OE
When a quadrilateral is inscribed in a circle, opposite angles must add up to 180°. This is because all angles of such a quadrilateral are inscribed angles of a circle. For example, angle A D C intercepts arc ABC, and angle ABC intercepts arc ADC.
The two arcs constitute the entire circle. Thus, the sum of the arcs intercepted by these angles is a whole circle, or 360, and the angles must sum to 1/2 (360°) = 180°. (This follows from the rule that the measure of an inscribed angle of a circle is one-half the measure of the corresponding central angle.) This gives x = 180 - 60 = 120 and y = 180 - 110 = 70. Combining, x - y = 120 - 70 = 50.