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Find the measure of angle ๐ด๐ต๐ถ.
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Letโs begin by identifying the angle whose measure weโre asked to find.
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Itโs the angle formed when we move from ๐ด to ๐ต to ๐ถ.
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So thatโs this angle here on the figure.
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Now this is an inscribed angle on the circleโs circumference.
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So we know that its measure will be one-half of its intercepted arc.
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Its intercepted arc is the arc ๐ด๐ถ.
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So we have the equation the measure of the angle ๐ด๐ต๐ถ is equal to one-half the measure of the arc ๐ด๐ถ.
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Letโs consider then how we might be able to calculate the measure of this arc.
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We can see that the other information given in the question is, firstly, the angle formed by the intersection of two chords inside a circle, the chords ๐ด๐ต and ๐ถ๐ท.
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Weโre also told the measure of the arc intercepted by this angle, the measure of the arc ๐ต๐ท, which is 98 degrees.
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The angles of intersecting chords theorem tells us that the measure of the angle between two chords that intersect inside a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
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The arc intercepted by the angle of 88 degrees is the arc ๐ต๐ท, and the arc intercepted by its vertical angle is the arc ๐ด๐ถ.
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So we can form an equation 88 degrees is equal to one-half the measure of the arc ๐ต๐ท plus the measure of the arc ๐ด๐ถ.
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Remember though that we know the measure of the arc ๐ต๐ท.
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Itโs given to us in the figure as 98 degrees.
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So we can substitute this value into our equation, and weโll then be able to solve to find the measure of the arc ๐ด๐ถ.
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We have 88 degrees is equal to one-half of 98 degrees plus the measure of the arc ๐ด๐ถ.
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Multiplying each side of the equation by two, we have 176 degrees equals 98 degrees plus the measure of the arc ๐ด๐ถ.
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And finally, subtracting 98 degrees from each side, we find that the measure of the arc ๐ด๐ถ is 78 degrees.
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The final step in this problem is to take this value for the measure of the arc ๐ด๐ถ and substitute it into our first equation.
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We have then that the measure of the angle ๐ด๐ต๐ถ, which is half the measure of its intercepted arc, is one-half multiplied by 78 degrees, which is 39 degrees.
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So, by recalling the relationship between the measures of an inscribed angle and its intercepted arc and also the angles of intersecting chords theorem, weโve found that the measure of the angle ๐ด๐ต๐ถ is 39 degrees.