Find The Sum Of Angles X and Y

By theorem.
$PT=PS$

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$\color{#D61F06}{\text{Length of tangents from external point are equal.}}$

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This makes $\triangle PTS$ isosceles .
Using Angle Sum Property of $\triangle$ .
$\Rightarrow y+y+30°=180°$
$\Rightarrow \boxed{y=75°}$
Now, by Alternate segment theorem
$\Rightarrow x=y=75°$

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$\color{#3D99F6}{\text{Angle between tangent and chord at the point of contact is equal to the angle in the alternate segment.}}$

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$\Rightarrow x+y=75°+75°=\boxed{150°}$

Note that $\Delta PTS$ is isosceles and thus we have $y+y+30^\circ=180^\circ \Rightarrow y=75^\circ$ . By Alternate segment theorem we have $x=y=75^\circ \rightarrow x+y=75^\circ+75^\circ=\boxed{150^\circ}$

In isosceles triangle who has angle 30, 2Y=150. In the circle X=Y because they measure with half of the same ark. So X+Y=2Y=150.

You don't even need to calculate the values of x and y. Notice that via the alternate segment theorem, the angle x is the same as the missing angle in the long triangle to the right. So the angles of the triangle are x, y, and 30. So the sum of x and y must be 180-30 =||150||

When two tangents are drawn from the same point i.e 30, they are the same length making it an isosceles triangle.

So y=(180-30)/2 =75

According to alternate segment theorem, x=y

So x=75 Therefore x+y=75 +75 =150 degrees