Answer:
The correct option is D.
Step-by-step explanation:
Let the center of the circle be O, ange YUX be x, angle ZXU be y and angle YOZ be z.
Given information: [tex]\angle XOU=38^{\circ}[/tex].
The Central Angle Theorem states that the central angle from two chosen points on the circle is always twice the inscribed angle from those two points.
Using Central Angle Theorem, we get
[tex]\angle UOZ=2y[/tex]
[tex]\angle XOY=2x[/tex]
The sum of central anges is 30 degree.
[tex]\angle XOY+\angle XOU+\angle UOZ+\angle YOZ=360^{\circ}[/tex]
[tex]2x+38^{\circ}+2y+z=360^{\circ}[/tex]
[tex]2(x+y)+38^{\circ}+z=360^{\circ}[/tex] .... (1)
In triangle XUV, the measure of exterior angle XVY is 90 degree.
According to the exterior angle theorem of triangles, the sum two interior angles is equal to the opposite exterior angle. By using exterior angle theorem of triangles, we get
[tex]x+y=90[/tex]
Put this value in equation 1.
[tex]2(90)^{\circ}+38^{\circ}+z=360^{\circ}[/tex]
[tex]180^{\circ}+38^{\circ}+z=360^{\circ}[/tex]
[tex]218^{\circ}+z=360^{\circ}[/tex]
[tex]z=360^{\circ}-218^{\circ}[/tex]
[tex]z=142^{\circ}[/tex]
Therefore option D is correct.
