Video: Finding the Measure of an Angle given the Measure of an Angle of Tangency by Using the Properties of Tangents to the Circle

Given that 𝑚∠𝑍𝑌𝐿 = 122°, find 𝑚∠𝑋.

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Video Transcript

Given that the measure of angle 𝑍𝑌𝐿 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 𝑌. We also have a chord drawn on the circle from point 𝑍 to point 𝑌. We are told that the measure of angle 𝑍𝑌𝐿 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 𝑋𝑌. Triangle 𝑋𝑌𝑍 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 𝑋𝑌𝑍 is equal to angle 𝑋𝑍𝑌.

Angles on a straight line sum to 180 degrees. This means that we can calculate the measure of angle 𝑋𝑌𝑍 by subtracting 122 from 180. This is equal to 58 degrees. Angles 𝑋𝑌𝑍 and 𝑋𝑍𝑌 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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