4. Determine the measure of the acute angle between the diagonals $\overline{A D}$ and $\overline{C G}$ in a regular octagon ABCDEFGH.
First solution. Let $S$ be the center of the circumscribed circle of the regular octagon $ABCDEFGH$.

![](https://cdn.mathpix.com/cropped/2024_05_30_1ae5e90785d026149547g-30.jpg?height=680&width=688&top_left_y=1559&top_left_x=247)

Triangle $\triangle C S D$ is isosceles and $|\angle C S D|=\frac{1}{8} \cdot 360^{\circ}=45^{\circ}$.

Therefore, $|\angle S C D|=|\angle S D C|=67.5^{\circ}$.

Triangle $\triangle A S D$ is isosceles and $|\angle A S D|=\frac{3}{8} \cdot 360^{\circ}=135^{\circ}$.

Therefore, $|\angle S A D|=|\angle S D A|=22.5^{\circ}$.

We have $|\angle A D C|=|\angle S D C|-|\angle S D A|=67.5^{\circ}-22.5^{\circ}=45^{\circ}$.

1 POINT

Finally, using the fact that the sum of the angles in a triangle is $180^{\circ}$, we calculate the measure of the desired angle as $180^{\circ}-|\angle A D C|-|\angle S C D|=180^{\circ}-45^{\circ}-67.5^{\circ}=67.5^{\circ}$.

1 POINT

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