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?

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

• Have you tried a smaller case, such as with $3$ straight lines? You can group them by the 'size' of the angle: $1$ angle and $2$ consecutive angles. What happens when you have $3$ consecutive angles? Jul 22, 2019 at 8:58

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 $$180$$.

• Did you mean $20$ rays? Jul 22, 2019 at 9:11
• They can be an array of 20 lines around the center, or 20 rays... Isn't it just the same? Jul 22, 2019 at 9:22

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

• This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review Jul 22, 2019 at 9:59
• @nmasanta I think hints are fine, since the question says "I have no idea where to start" and this gives a place to start. Pursuing it correctly also gives a relatively simple solution. Jul 22, 2019 at 11:25
• @nmasanta yes, it does provide an idea where to start, as asked. Did you want a full solution ? Jul 22, 2019 at 11:26
• But it's not an answer, you can give your valuable hints in the comment portion too. Am I right ? Jul 22, 2019 at 11:36
• @nmasanta You can give hints in comments it is true, but a question can ask for help, in which case the answer is to give help rather than showing a full solution. You will note that the full answer given does not use permutations/combinations as mentioned in the question, whereas I indicate how they can be used. Feel free to disagree, but I was keen for OP to try to make progress themselves. I agree this is not an answer to "how many angles", but it is an answer to "help me get started". Jul 22, 2019 at 12:05