$\text{There are angles} \angle AOB ,\angle BOC, \angle COD \text{on a plane}.$

Given that $OM,ON$ bisect $\angle AOB,\angle COD$ respectively, And that $\angle MON=50^\circ ,\angle BOC=10^\circ$, as in the figure, find $\angle AOD.$

But there was NO picture! And I was thinking about how to solve it but realised the possibilities are too complicated, as the size of angle AOB may vary.

Given that every angle is smaller than 180 degrees and bigger than 0 degrees, how should I separate the possibilities?

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TopNewest@Frisk Dreemurr

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@Jeff Giff, are you there?

I have got a solution for your question, but it'll take some seconds to send, alright...?

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When I read this question, I decided to make a sketch of my own. So my first version was this:

(angles and lengths both aren't to scale)$50°$ is visibly the sum of the bisector of AOB, the whole BOC, and the bisector of COD

Subtracting only BOC to get the bisectors give $50° - 10° = 40°$, which is the sum of BOM and CON

Multiplying it by two should give the sum of angles AOB and COD, which is $80°$

To get the value of AOD, we must add AOB, BOC and COD, which is $80° + 10° = 90°$

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thanks! That was part of my solution, but AOB might be bigger than 100 degrees, so MON could be inside AOB😅 not to mention that OC could be inside AOB, giving multiple answers😊

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But N must be part of ON, and according to the question, ON is the bisector of COD

Same goes to M and AOB

So AOB can not be bigger than $40°$

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As long as it is good, I'd love to see your solution :)

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Thanks a lot anyways, I hope to mention some more people but it turns out I can’t :P

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@Jeff Giff

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Umm I should get to send the solution at the weekend, my maths teacher says I’m wrong and I still don’t know exactly why 😊

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I'll ask my maths teacher to solve it.

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Geez guess what my maths teacher accidentally solved it wrong mwah hah hah :D

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So I’m correct :P and I’m posting the solution

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First, let $\angle CON=x^\circ$.

## Case 1

When $0^\circ \le \angle AOB \le 10^\circ$ so $\angle AOB$ is in $\angle BOC_1$, as in the figure, there are 4 possibilities of AOD(the number after D shows the C and N it’s contributed by, e.g. $AOD_{11}$ matches with $C_1$ & $N_1$):

$\begin{aligned} \angle AOD_{11}&=\angle C_1OD_{11}-(\angle BOC_1-\angle AOB)\\ &=\angle C_1OD_{11}+\angle AOB-\angle BOC_1 \\ &=2x^\circ +2\angle MOB-10^\circ\\ &=(2x-10)^\circ +2(\angle BOC_1+\angle MON_1-\angle C_1ON_1)\\ &=(2x-10)^\circ +2(60-x)^\circ =\red{110^\circ};\\ \angle AOD_{12}&=\angle BOC_1+\angle C_1OD_{12}-\angle AOB\\ &=(10+2x)^\circ -2[10^\circ -(50-x)^\circ ]=\red{90^\circ};\\ \angle AOD_{21}&=\angle AOB+\angle BOC_2+\angle C_2OD_{21}\\ &=2(\angle MOB+\angle C_2ON_1)+10^\circ \\ &=2(\angle MON_1-\angle BOC_2)+10^\circ = 2\times (50-10)^\circ +10^\circ =2\times 40^\circ +10 ^\circ =\red{90^\circ} ; \\ \angle AOD_{22}&=C_2OD_{22}-\angle AOB-\angle BOC_2\\ &=2x^\circ -2\angle MOB-10^\circ \\ &=(2x-10)^\circ -2(x-50-10)^\circ =\red{110^\circ}. \end{aligned}$ -NOT FINISHED YET-

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@Valentin Duringer here is a problem that can be included in dynamic geometry, because this is mad as the picture is not given, meaning we have to think of all 6 varieties of AOB and result in 20 AODs, which only return two values! :D It was my HOMEWORK and my maths teacher nearly said holy —— seeing my solution - didn’t realise what chaos would be caused without the picture :D

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hum..chaotic indeed -)

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I’ll give the latex of the problem here later, copy it if you have time :)

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CAREFULLY READ ASSUMPTIONS BEFORE CALCULATINGThere are rods OA, OB, OC, OD(not necessarily ordered) on a plane. Given two rods OM, ON that bisect \ ( \angle AOB \ ) and \ ( \angle COD\ ) respectively, and that \ ( \angle MON =50^\circ ,\angle BOC =10^\circ \ ) , while the value of \ ( \angle AOB \ ) varies, find \ ( \angle AOD. \ )

Assumptions:All angles are in range (0,\pi). That is to assure which side the angle bisector is on.

The size of AOB varies and so does the size of AOD. You will find a few finite possibilities for AOD. Input the product of all possible answers IN DEGREES as result.

e.g. if possible results are 10 degrees ,20 degrees and 30 degrees, then input 10\times 20\times 30 =6000 as your answer.

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Just remember to add the latex brackets :)

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