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Question Video: Finding the Measure of a Central Angle given Its Arcโ€™s Measure Using Another Inscribed Angleโ€™s Measure Mathematics

Find ๐‘šโˆ ๐ด๐ต๐ถ.

02:55

Video Transcript

Find the measure of angle ๐ด๐ต๐ถ.

Letโ€™s begin by identifying the angle whose measure weโ€™re asked to find. Itโ€™s the angle formed when we move from ๐ด to ๐ต to ๐ถ. So thatโ€™s this angle here on the figure. Now this is an inscribed angle on the circleโ€™s circumference. So we know that its measure will be one-half of its intercepted arc. Its intercepted arc is the arc ๐ด๐ถ. So we have the equation the measure of the angle ๐ด๐ต๐ถ is equal to one-half the measure of the arc ๐ด๐ถ.

Letโ€™s consider then how we might be able to calculate the measure of this arc. 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 ๐ด๐ต and ๐ถ๐ท. Weโ€™re also told the measure of the arc intercepted by this angle, the measure of the arc ๐ต๐ท, which is 98 degrees. 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. The arc intercepted by the angle of 88 degrees is the arc ๐ต๐ท, and the arc intercepted by its vertical angle is the arc ๐ด๐ถ. So we can form an equation 88 degrees is equal to one-half the measure of the arc ๐ต๐ท plus the measure of the arc ๐ด๐ถ.

Remember though that we know the measure of the arc ๐ต๐ท. Itโ€™s given to us in the figure as 98 degrees. So we can substitute this value into our equation, and weโ€™ll then be able to solve to find the measure of the arc ๐ด๐ถ. We have 88 degrees is equal to one-half of 98 degrees plus the measure of the arc ๐ด๐ถ. Multiplying each side of the equation by two, we have 176 degrees equals 98 degrees plus the measure of the arc ๐ด๐ถ. And finally, subtracting 98 degrees from each side, we find that the measure of the arc ๐ด๐ถ is 78 degrees.

The final step in this problem is to take this value for the measure of the arc ๐ด๐ถ and substitute it into our first equation. We have then that the measure of the angle ๐ด๐ต๐ถ, which is half the measure of its intercepted arc, is one-half multiplied by 78 degrees, which is 39 degrees. So, by recalling the relationship between the measures of an inscribed angle and its intercepted arc and also the angles of intersecting chords theorem, weโ€™ve found that the measure of the angle ๐ด๐ต๐ถ is 39 degrees.

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