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Question Video: Finding the Measure of an Angle Given Its Arcโ€™s Measure Using Another Inscribed Angle Mathematics • 11th Grade

Given that ๐‘šโˆ ๐ธ๐ถ๐ท = 54ยฐ and ๐‘šโˆ ๐น๐ต๐ท = 78ยฐ, find ๐‘ฅ and ๐‘ฆ.

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Video Transcript

Given that the measure of angle ๐ธ๐ถ๐ท equals 54 degrees and the measure of angle ๐น๐ต๐ท equals 78 degrees, find ๐‘ฅ and ๐‘ฆ.

First, we can add the measure of the two angles weโ€™re given into the diagram. To solve further, we will need to remember the alternate segment theorem. If we have points ๐ด and ๐ต that fall on a circle and point ๐ถ is the point where a tangent passing through the line intersects the circle, here the measure of angle ๐ถ๐ต๐ด, weโ€™ve labeled as ๐œƒ, and that will be equal to the measure of angle ๐ท๐ถ๐ด. Weโ€™ll let the measure of angle ๐ถ๐ด๐ต be labeled as ๐›ฝ, and this angle will be equal to the measure of angle ๐ธ๐ถ๐ต. Using this theorem, we can find that the measure of angle ๐ท๐ต๐ถ equals 54 degrees. The measure of angle ๐ท๐ถ๐ต equals 78 degrees. And from there, weโ€™ll see the angle ๐ต๐ถ๐ด and angle ๐ถ๐ต๐ด both measure ๐‘ฅ degrees.

Because ๐ต๐ถ๐ท forms a triangle, we know that the three angles inside sum to 180 degrees. And therefore, we can find ๐‘ฅ by taking 180 and subtracting the other two angles 54 plus 78. And we see that ๐‘ฅ equals 48. To find ๐‘ฆ, weโ€™ll consider the triangle ๐ด๐ต๐ถ. Itโ€™s made up of the angles ๐‘ฅ, ๐‘ฅ, and ๐‘ฆ. So ๐‘ฅ plus ๐‘ฅ plus ๐‘ฆ must equal 180 degrees. Weโ€™ll plug in what we know. 48 plus 48 is 96. So we subtract 96 from both sides. And we find that ๐‘ฆ must be equal to 84.

Our missing ๐‘ฅ- and ๐‘ฆ-values are 48 and 84, respectively. And we found this using the alternate segment theorem.

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