# Sum of angles under which a fixed line segment is seen from points situated on another line segment

I have a question, like a picture attached below. I can find each angle by sin( or cosine) rule, but I think there is an easy way...a clue ...a concept which made it easy. can someone help me? I do appreciate any hint.

for example to find $$A$$ I use $$BC=\sqrt 2, AC=6 , AB=\sqrt {26}\\\cos(A)=\frac{c^2+b^2-a^2}{2bc}=\frac{36+26-2}{2*6*\sqrt{26}}$$ then find $$A=11.3099$$ and do like this for all the angles. But it is not the satisfying method. ( the gray squares are equal) Thanks in advanced.

• As segments cut parallel lines, you either add them up at point $B$ or at point $C$. That I think gives you $90^\circ$. Commented Oct 16, 2021 at 18:40
• Where did this problem come from? (In particular, the image.)
– anon
Commented Oct 16, 2021 at 18:41
• Please, could you answer the question of runway44 ? Commented Oct 18, 2021 at 17:09
• A question with a similar solution here. This question has motivated my proposal of a new title for your question. Commented Oct 18, 2021 at 17:16

The equal angles in the solution come from the fact that $$BCDE, BCEF, BCFG, BCGH, BCHA, BIAJ$$ are all parallelograms. This is because $$BC,ED$$ have the same slope and so do $$BE, CD$$, etc. Then, $$\angle BDC=\angle DBE$$ etc. by alternate angles. All the angles put together add up to rotating $$BD$$ onto $$BJ$$. Since $$BJ$$ is vertical and $$BD$$ is horizontal, they then add to $$90$$ degrees.

(Thanks @ACB for tidying up the image)

• @ACB Thanks so much! Commented Oct 17, 2021 at 15:41
• Very fancy and lovely answer! Commented Oct 18, 2021 at 17:21
• @Nightflight Thanks! Commented Oct 18, 2021 at 21:52
• Fantastic mathematical insight! Commented Jun 22, 2022 at 17:29

Hint 1:

The twelve line segments come in six parallel pairs.

Hint 2:

Move the angles around so that the parallel lines match up.

Solution:

Excuse my poor paint skills.

• thank you, but can you clarify a bit more? please Commented Oct 16, 2021 at 18:35
• @Khosrotash I've added the corresponding picture. Commented Oct 16, 2021 at 18:36
• Thank you I got the idea. Commented Oct 17, 2021 at 4:54