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.