9.2. Point $B$ is the midpoint of segment $A C$. Square $A B D E$ and equilateral triangle $B C F$ are located in the same half-plane relative to line $A C$. Find (in degrees) the measure of the acute angle between lines $C D$ and $A F$.
Answer: $75^{\circ}$.

Solution. Note that points $A D F C$ lie on the same circle with center at point $B$. The arc $F C$ is $60^{\circ}$. Therefore, the angle $F A C$ is $30^{\circ}$ (by the inscribed angle theorem). Triangle $B D C$ is isosceles with a right angle at vertex $B$. Therefore, the angle $D C A$ is $\pi / 4$. Thus, the acute angle between the lines $A F$ and $D C$ is $30^{\circ}+45^{\circ}=75^{\circ}$.

Criteria.

(-.) Progress in the problem (finding many equal segments and angles) without obtaining the answer.

(+/2) Correct answer, significant gaps in justification (but without logical errors),

or errors that did not affect the answer.

(+.) Insufficient justification of some (not difficult) facts or omission of some more or less obvious steps, or the obtuse angle was calculated instead of the acute one, or a minor arithmetic error that did not affect the answer.

(+) Correct answer with full justification and no errors in the solution.