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Question Video: Finding the Measure of an Angle given the Measure of an Angle of Tangency by Using the Properties of Tangents to the Circle Mathematics • 11th Grade

Given that π‘šβˆ π‘π‘ŒπΏ = 122Β°, find π‘šβˆ π‘‹.

02:05

Video Transcript

Given that the measure of angle π‘π‘ŒπΏ is equal to 122 degrees, find the measure of angle 𝑋.

In our diagram, we have two tangents starting from the point 𝑋. The top one touches the circle at point 𝑍 and the bottom one touches the circle at point π‘Œ. We also have a chord drawn on the circle from point 𝑍 to point π‘Œ. We are told that the measure of angle π‘π‘ŒπΏ is 122 degrees. We recall that two tangents drawn from the same point must be equal in length. This means that the line segment 𝑋𝑍 is equal to the line segment π‘‹π‘Œ. Triangle π‘‹π‘Œπ‘ is therefore isosceles, as it has two equal length sides. In any isosceles triangle, the measure of two angles are equal. In this case, angle π‘‹π‘Œπ‘ is equal to angle π‘‹π‘π‘Œ.

Angles on a straight line sum to 180 degrees. This means that we can calculate the measure of angle π‘‹π‘Œπ‘ by subtracting 122 from 180. This is equal to 58 degrees. Angles π‘‹π‘Œπ‘ and π‘‹π‘π‘Œ are both equal to 58 degrees. Our aim in this question is to find the measure of angle 𝑋, and we know that angles in a triangle also sum to 180 degrees. Angle 𝑋 is therefore equal to 180 minus 58 plus 58. 58 plus 58 is equal to 116, and subtracting this from 180 gives us 64. The measure of angle 𝑋 is therefore equal to 64 degrees.

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