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Problem 11. The figure below shows two congruent circles centered at A and B. If ∠P F E = 75◦ , then what is the measure of angle EP F ? E A P B F Solution. The measure of angle EP F is 10◦ . Draw segments AB and BE and note that the former is a radius in circle A and that both are radii in circle B. Let the measure of angle EP F = x. Then because AP = AB, we find that ∠ABP = x. Since ∠BAE is an exterior angle to triangle ABP , we have ∠BEA = ∠AP B + ∠ABP = 2x. and it then follows that also ∠AEB = 2x. Next note that because EB = F B, we have ∠BEF = ∠AF E = 75◦ , and hence ∠EBF = 30◦ . Because ∠EBF is an exterior angle to triangle BEP , we have 30◦ = ∠EBF = ∠BEP + ∠BP E = 2x + x = 3x, so 10◦ . E 2x A 75 2x P x x 30 B 75 F