In the figure, $PA$ is tangent to semicircle $SAR$, $PB$ is tangent to semicircle $RBT$, and $SRT$ is a straight line. If arc $AS$ is $58^\circ$ and arc $BT$ is $37^\circ$, then find $\angle APB$, in degrees.
+130536 Draw AR
Since arc AS is 58°....then angle ARS also = 58°
Similarly...draw RB
And since arc BT is 37°, then angle BRT also = 37°
So...we have quadrilateral PARB
Since angle SRT = 180° ....then angle ARB =
180 - angle ARS - angle BRT =
180 - 58 - 37 = 85°
And angles RAP and RBP = 90°
And the sum of the interior angles of a quadrilateral = 360°
So
angle ARB + angle RAP + angle RBP + angle APB = 360
85 + 90 + 90 + angle APB = 360
265 + APB = 360
APB = 95°
