Question

If W is the center of the circle below, and arc XY measures 52, what is the measure of angle xyz , in degrees?

Answer 1

angle XWZ = 52°

Assuming YZ is a diameter (since it passes thru the center W):

Angle XWY = 180° - 52° = 128° [because of the adjacent angles rule]

Now WX = WY = radius of circle

So ΔWXY is isosceles

So angle WXY = angle WYX = (1/2) [180° - 128°] = 26°

and angle XYW = angle XYZ [actually the same angle]

so angle XYZ = 26°

Answer 2

Don't you mean what is the measure of angle XZY?

Answer 3

Angle XZY has the arc length 52 so knowing the relationship of an inscribed angle with its vertex on the circles outline you can find the angle easily
\[\theta = \frac{ 1 }{ 2} arc\]