Problem 2. Given points $A, B, C$, and $D$, none of which are collinear. List all the angles whose vertex is one of the given points and whose sides include two of the remaining three points. Then, list all the line segments formed by these 4 points. Which are there more of: line segments or angles?

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Given points $A, B, C$, and $D$, none of which are collinear. 

1. List all the angles whose vertex is one of the given points and whose sides include two of the remaining three points.
2. List all the line segments formed by these 4 points.
3. Determine whether there are more line segments or angles.
Solution. There are 12 angles:

$\measuredangle A B C, \measuredangle A B D, \measuredangle D B C, \measuredangle B C D, \measuredangle B C A, \measuredangle A C D$,

$\measuredangle C D A, \measuredangle C D B, \measuredangle B D A, \measuredangle D A B, \measuredangle D A C, \measuredangle C A B$ There are 6 segments: $A B, A C, A D, B C, B D, C D$. Therefore, there are more angles.

![](https://cdn.mathpix.com/cropped/2024_06_05_1cb7beb611b796adc950g-03.jpg?height=290&width=345&top_left_y=463&top_left_x=1123)