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Geometry Difficulty 4.7 AIME Prove it Italy

Problem:

A square-based pyramid has base edge length $\sqrt{3}$ and all the edges of the lateral faces are of length $\sqrt{2}$ . How many degrees does the angle between two edges not belonging to the same lateral face measure?

Solution

Solution:

The answer is $120$ . The required angle is the angle at the vertex of the isosceles triangle whose base is the diagonal of the base and whose slanted sides are two edges of the lateral faces. The base measures $\sqrt{6}$ and the ratio between the slanted side and half the base is $\frac{\sqrt{3}}{2}$ : this is the sine of half the required angle. The measure of the angle is $120^{\circ}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.