Question

How do I find x ??????

Answer 1

Consult the attached image.
Let O be the center of the circle. Let A and B be points on the line tangent to the circle, but not on the circle itself. Let P be the point of tangency where the tangent line intersects the circle. Let PQ be a chord of the circle, and let C be a point on the line containing the chord PQ.


The central angle subtended by the arc with measure 220 degrees also has a measure of 220 degrees, which means angle POQ has measure 360 - 220 = 140 degrees.

Triangle OPQ is isosceles, which means angles and OPQ and OQP are congruent. The measures of the interior angles of any triangle must add to 180 degrees. So if
`y=m\angle OPQ=m\angle OQP`, then


`180^\circ=2y+140^\circ\implies y=20^\circ`

Finally, angles APC and OPQ are complementary because the line segment OP is perpendicular to the tangent line AB. So


`x+y=90^\circ\implies x=70^\circ`