# Question Video: Finding the Measure of an Angle Given Its Arcβs Measure Using Another Inscribed Angle Mathematics • 11th Grade

Given that πβ πΈπΆπ· = 54Β° and πβ πΉπ΅π· = 78Β°, find π₯ and π¦.

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### Video Transcript

Given that the measure of angle πΈπΆπ· equals 54 degrees and the measure of angle πΉπ΅π· equals 78 degrees, find π₯ and π¦.

First, we can add the measure of the two angles weβre given into the diagram. To solve further, we will need to remember the alternate segment theorem. If we have points π΄ and π΅ that fall on a circle and point πΆ is the point where a tangent passing through the line intersects the circle, here the measure of angle πΆπ΅π΄, weβve labeled as π, and that will be equal to the measure of angle π·πΆπ΄. Weβll let the measure of angle πΆπ΄π΅ be labeled as π½, and this angle will be equal to the measure of angle πΈπΆπ΅. Using this theorem, we can find that the measure of angle π·π΅πΆ equals 54 degrees. The measure of angle π·πΆπ΅ equals 78 degrees. And from there, weβll see the angle π΅πΆπ΄ and angle πΆπ΅π΄ both measure π₯ degrees.

Because π΅πΆπ· forms a triangle, we know that the three angles inside sum to 180 degrees. And therefore, we can find π₯ by taking 180 and subtracting the other two angles 54 plus 78. And we see that π₯ equals 48. To find π¦, weβll consider the triangle π΄π΅πΆ. Itβs made up of the angles π₯, π₯, and π¦. So π₯ plus π₯ plus π¦ must equal 180 degrees. Weβll plug in what we know. 48 plus 48 is 96. So we subtract 96 from both sides. And we find that π¦ must be equal to 84.

Our missing π₯- and π¦-values are 48 and 84, respectively. And we found this using the alternate segment theorem.