In the figure, if \(FR=FG\), the measure of arc \(FG\) is \(130^{\circ}\), and measure of arc \(FQ\) is \(28^{\circ}\), then what is \(\angle RPG\), in degrees?
+658 1. Connect \(RG\)
2. Since \(FR=FG\), triangle \(FRG\) is isosceles with \(F\) as the vertex.
3. Angle \(GRF\) subtends arc \(GF\), which is 130, therefore angle \(GRF\) = 130/2 = \(65\)
4. Since \(FRG\) is isosceles, that means \(FRG=RGF\).
5. Angle \(GFR\) = \(50\) because all triangles have an angle sum of 180, and we know that two angles are 65 degrees. (180 - 65 * 2)
6. Angle \(PGF\) subtends arc \(FQ\), so angle \(PGF\) = 28/2 = \(14\)
7. Conside triangle \(PFG\), we know the measure of two angles \(GFR\) and \(PGF\) is \(50\) and \(14\), respectively. That means the third angle \(GPF\) is 180 - 50 - 14 = \(116\)
8. By supplements angle \(RPG=180-116 = 64\)
Done
