Answers:
x = 66
y = 66
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Explanation
Triangle ABO and triangle ACO are right triangles. This is because the tangent segments form 90 degree angles with the radius at the point of tangency. In other words: angle ABO = 90 and angle ACO = 90.
Since triangle ABO is a right triangle, angle AOB = 90 - (angleBAO) = 90-24 = 66 degrees.
[tex]\triangle ABO \cong \triangle ACO[/tex] due to the Hypotenuse Leg (HL) theorem. The two triangles are mirror copies of each other.
Therefore, x = 66 because this angle is a mirror copy of angle AOB.
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Central angle COB = 2x = 2*66 = 132 degrees which leads to minor arc BC having this measure as well.
Half of which is the the inscribed angle that subtends this arc.
Inscribed angle BDC = (1/2)*(arc BC) = (1/2)*(132) = 66 is the value of y.
It's not a typo that both x and y have the same value.
It turns out that y = x even if we changed that "24" to some other value.
