# Langley's adventitious angles

Langley's adventitious angles is a seemingly simple problem first posed in 1922 by E. M. Langley in connection
with an isosceles triangle. In its original
form, it is stated as follows: *ABC* is an isosceles triangle. *B* = *C* = 80°. *CF* at 30° to *AC* cuts *AB* in *F*. *BE* at 20° to *AB* cuts *AC* in *E*. Prove angle *BEF* = 30°. (No mention is made of *D*.
Perhaps it is at the intersection of *BE* and *CF*.)

A number of solutions appeared shortly after, including this one given by
J. W. Mercer: Draw *BG* at 20° to *BC*, cutting *CA* in *G*. Then angle *GBF* = 60° and angles *BGC* and *BCG* are 80°. So *BC* = *BG*. Also angle *BCF* = angle *BFC* = 50°, so *BF* = *BC* and triangle *BFG* is equilateral. But angle *GBE* = 40° = angle *BEG*,
so *BG* = *GE* = *GF*. And angle ° = 40°, hence *GEF* = 70° and *BEF* = 30°.

### Reference

1. Langley, E. M. "Problem 644." *Mathematical Gazette*, 11: 173,
1922.