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Geometry

# Alternate Segment Theorem

In the above diagram, $$\overline{ PAT}$$ is tangent to circle $$O$$ at point $$A.$$ If $\angle BCA={56}^\circ \text{ and } \angle CAP={49}^\circ,$ what are the measures of the angles $$\angle x$$ and $$\angle y ?$$

Note: The above diagram is not drawn to scale.

If $$\overline{AT}$$ is tangent to circle $$O$$ with radius $$4$$ at $$A$$ and $$\angle BAT=60^\circ,$$ what is the area of $$\triangle OAB ?$$

In the above diagram, if $$\overline{PTQ}$$ is tangent to both circles at $$T$$ and $\angle ABT={72}^\circ, \angle DCT={76}^\circ,$ what is the value of $$\angle x+\angle y$$ in degrees?

If $$\overline{PA}$$ and $$\overline{PB}$$ are tangent to circle $$O$$ at points $$A$$ and $$B,$$ respectively, and $$\angle APB={68}^\circ,$$ what is the measure of $$\angle x$$ in degrees?

Note: The above diagram is not drawn to scale.

In the above diagram, quadrilateral $$ABCD$$ is inscribed in circle $$O$$ and $$\overline{AT}$$ is tangent to circle $$O$$ at point $$A.$$ If $$\angle BAT={59}^\circ$$ and $$\angle DBA={39}^\circ,$$ what is the measures of $$\angle x$$ in degrees?

Note: The above diagram is not drawn to scale.

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