To calculate the measure of angle XY, we draw a line from Y to the circle's center, O, and label the intersection.Since XYO and OXY are supplementary angles, summing to 180 degrees, angle XYQ is an exterior angle of XYZ, and XYQ = XYZ + YXZ = 106 + 42 = 148 °.
The measure of ∠XY is 148° .
This can be found using the following steps:
Draw a line segment from Y to the center of the circle, O.
Label the intersection of this line segment and the circle as Q.
Since
OQ is a radius of the circle, it follows that ∠QYO=90°..
Since XY is tangent to the circle at Y, it follows that ∠YXQ=90°.
Since ∠QYO and ∠YXQ are complementary angles, it follows that they have equal measures, which is 90 °
Since ∠OXY=∠QYO+∠YXQ, it follows that ∠OXY=90° +90 ° =180 °.
Since ∠XYO and ∠OXY are supplementary angles, it follows that they have equal measures, which is 180° .
Since ∠XZY and ∠XYO are vertical angles, it follows that they have equal measures, which is 180°.
Since ∠XYZ and ∠XZY are supplementary angles, it follows that they have equal measures, which is 180 °.
Since ∠XYZ=180° −74° , it follows that ∠XYZ=106 °.
Since ∠XYQ is an exterior angle of △XYZ, it follows that ∠XYQ=∠XYZ+∠YXZ=106° +42°= 148°
Therefore, the measure of ∠XY is 148°.