The portal has been deactivated. Please contact your portal admin.

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.


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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.