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