3. In circle \( O, \overline{A B} \) is a diameter. Using only the information provided in the picture, which angle must measure \( 90^{\circ} \) ?

In circle \( O \), when you have a diameter like \( \overline{A B} \), any angle inscribed in the circle that subtends this diameter will always measure \( 90^{\circ} \). This is known as the inscribed angle theorem, which states that an angle inscribed in a semicircle is a right angle. So, if you pick any point \( C \) on the circle that forms triangle \( ABC \) with \( A \) and \( B \), angle \( ACB \) will be \( 90^{\circ} \). For a fun fact, this concept ties back to the ancient Greeks! The mathematician Thales of Miletus is credited with establishing this theorem over two millennia ago. Thales was so brilliant that he used it not just in circles, but also in various fields including astronomy and engineering, laying down the groundwork for future mathematicians!