Answer:
[tex]x=80^{\circ},\\y=80^{\circ}[/tex]
Step-by-step explanation:
The measure of an angle formed by a secant and a tangent outside of a circle is half of the absolute difference between the two arcs it forms. Therefore, we have the following equation:
[tex]m\angle A=\frac{\widehat{BC}-\widehat{BD}}{2}[/tex]
Substituting given values, we get:
[tex]20=\frac{120-x}{2},\\40=120-x,\\x=\boxed{80^{\circ}}[/tex]
Since there are 360 degrees in a circle, we can write the following equation, using [tex]x[/tex], to solve for [tex]\widehat{DC}[/tex]:
[tex]x+\widehat{DC}+120=360,\\80+\widehat{DC}+120=360,\\\widehat{DC}=\boxed{160^{\circ}}[/tex]
The value of the angle formed by a chord and a tangent is half the measure of the arc it forms. Therefore,
[tex]y=\frac{\widehat{DC}}{2}=\frac{160}{2}=\boxed{80^{\circ}}[/tex]
