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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?

Apr 28, 2020
edited by Guest  Apr 28, 2020

#1
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Where is the referred to figure ?

Apr 28, 2020
#2
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here's the diagram

Apr 28, 2020
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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

Apr 29, 2020