How many angles less than 180$^\circ$ can be formed by 10 straight lines which meet at a point, If no two of them are in the same straight line?

Question

I just have no idea where to start. The topic i'm studying currently is all about combinations and permutations.

Answer 1

First, think of the condition: Angles less than $180^o$ and no two lines are in the same line. With those conditions, the easiest representation would be to imagine you are cutting a circular cake. You need to slice it 10 times. Then, you will notice that you made an array of 20 lines from the center of the cake.

Here comes the problem. How can you count the angles less than $180^o$?

Let's name the points with letters, with center as $A$, the topmost point as $B$, then run clockwise and name it from $C$ to $U$. You will notice that for every point in the edge, with your angle running clockwise in increase (starting from $BAC$ to $BAK$ are all less than $180^o$. Any further than that, then the angle becomes greater than or equal to $180^o$. So, from $B$, you got 9 angles, moving to $C$, you'll get exactly 9 angles again. repeat it over and over and at $U$, you will still get 9 angles.

In short, the total number of angles less than $180^o$ is just $(9)(20)$ or $180$.

Answer 2

Hint: What happens with a pair of lines? Can you count such pairs in a more complex configuration?