Angles in Geometry done correctly

by kamatadu, Nov 2, 2024, 7:35 PM

POST BY AVIGYAN

This is going to be a very short blog-post discussing about the notations of angles.

Backstory: This blog-post is written because the notation $\measuredangle (\ell_1, \ell_2)$ confused me so much that I had to ask what it really means in an MSE post. I was still not satisfied with the explanation I got there. As I read a few more handouts, the notations became a bit more clear to me. I hope the idea I am sharing in this blog-post is indeed the correct one.

We start with vanilla angles first. Let's say we have three points $A$ , $B$ and $C$ . Then suppose we want to find the value of the angle $\angle ABC$ . We can extend the lines $AB$ and $BC$ to note that there are four angles that are formed. Which one do we take?

$[asy] /* Converted from GeoGebra by User:Azjps using Evan's magic cleaner https://github.com/vEnhance/dotfiles/blob/main/py-scripts/export-ggb-clean-asy.py */ /* A few re-additions are done using bubu-asy.py. This adds the dps, xmin, linewidth, fontsize and directions. https://github.com/Bubu-Droid/dotfiles/blob/master/bubu-scripts/bubu-asy.py */ pair A = (31.64835,16.28958); pair B = (40.11708,2.37181); pair C = (44.07794,21.53331); import graph; size(8cm); pen dps = linewidth(0.5) + fontsize(13); defaultpen(dps); real xmin = -5, xmax = 5, ymin = -5, ymax = 5; draw(arc(B,4.60985,78.32093,121.31981)--B--cycle, linewidth(0.6) + blue); draw(arc(B,4.60985,-101.67906,-58.68018)--B--cycle, linewidth(0.6) + blue); draw(arc(B,2.56591,-58.68018,78.32093)--B--cycle, linewidth(0.6) + red); draw(arc(B,2.56591,121.31981,258.32093)--B--cycle, linewidth(0.6) + red); draw(A--B, linewidth(0.6)); draw(B--C, linewidth(0.6)); draw((37.77614,-8.95301)--B, linewidth(0.6) + linetype("4 4")); draw(B--(45.63427,-6.69530), linewidth(0.6) + linetype("4 4")); draw(arc(B,4.60985,78.32093,121.31981), linewidth(0.6) + blue); draw(arc(B,4.27057,78.32093,121.31981), linewidth(0.6) + blue); draw(arc(B,4.60985,-101.67906,-58.68018), linewidth(0.6) + blue); draw(arc(B,4.27057,-101.67906,-58.68018), linewidth(0.6) + blue); dot("$A$", A, NW); dot("$B$", B, dir(180)); dot("$C$", C, N); [/asy]$

So here, we define the notation $\angle ABC$ as the magnitude of the angle with measure $\le 90^{\circ}$ angle of the four angles we found earlier. For this case, it is the blue angle.

But what does the notation $\angle (\ell_1,\ell_2)$ mean? Is it the one which is $\le 90^{\circ}$ or the other one? Does orientation matter?

This is where the doubt first comes up. In vanilla angles however, this question makes no sense as we always define the vanilla angle between two lines as the angle which is $\le 90^{\circ}$ . Vanilla angles do not care about orientation too.

Thus we can now safely conclude that $\angle (\ell_1,\ell_2) = \angle (\ell_2,\ell_1)$ . So far so good.

Before we begin, we should keep in mind that "Anti-clockwise is positive and clockwise is negative".

Let us now move onto directed angles. This is where things start behaving differently because suddenly, reflexive angles start being considered modulo $180^{\circ}$ and orientation also comes into play.

We begin with the same configuration. Let's say we have three points $A$ , $B$ and $C$ (where $A$ , $B$ and $C$ are in anti-clockwise order). Now we want to find the value of the directed angle $\measuredangle ABC$ .

$[asy] /* Converted from GeoGebra by User:Azjps using Evan's magic cleaner https://github.com/vEnhance/dotfiles/blob/main/py-scripts/export-ggb-clean-asy.py */ /* A few re-additions are done using bubu-asy.py. This adds the dps, xmin, linewidth, fontsize and directions. https://github.com/Bubu-Droid/dotfiles/blob/master/bubu-scripts/bubu-asy.py */ pair A = (31.64835,16.28958); pair B = (40.11708,2.37181); pair C = (44.07794,21.53331); import graph; size(8cm); pen dps = linewidth(0.5) + fontsize(13); defaultpen(dps); real xmin = -5, xmax = 5, ymin = -5, ymax = 5; draw(arc(B,4.60985,78.32093,121.31981)--B--cycle, linewidth(0.6) + blue); draw(arc(B,4.60985,-101.67906,-58.68018)--B--cycle, linewidth(0.6) + blue); draw(arc(B,2.56591,-58.68018,78.32093)--B--cycle, linewidth(0.6) + red); draw(arc(B,2.56591,121.31981,258.32093)--B--cycle, linewidth(0.6) + red); draw(A--B, linewidth(0.6)); draw(B--C, linewidth(0.6)); draw((37.77614,-8.95301)--B, linewidth(0.6) + linetype("4 4")); draw(B--(45.63427,-6.69530), linewidth(0.6) + linetype("4 4")); draw(arc(B,4.60985,78.32093,121.31981), linewidth(0.6) + blue,BeginArcArrow(6)); draw(arc(B,4.60985,-101.67906,-58.68018), linewidth(0.6) + blue); draw(arc(B,4.27057,-101.67906,-58.68018), linewidth(0.6) + blue); draw(arc(B,2.56591,121.31981,251.28453), linewidth(0.6) + red,EndArcArrow(6)); dot("$A$", A, NW); dot("$B$", B, dir(180)); dot("$C$", C, N); [/asy]$

For doing this, we take the line $BA$ and rotate it in an anti-clockwise direction w.r.t. $B$ until it gets super-imposed on the line $BC$ . (If you thought it was the other way around, you were wrong!)

The orientation does not depend on the ordering of the vertices but rather on the direction of rotation. (I had the opposite idea a few weeks back too!) This statement might feel a bit ambiguous right now, but just hold on.

So when we are finding the value of $\measuredangle ABC$ (as shown in the image above), the positive value of the directed angle is $> 90^{\circ}$ (the red angle) for the case in the image. Now if we take this value modulo $180^{\circ}$ to make the magnitude $\le 90^{\circ}$ , we find that the value is actually negative.

The anti-clockwise ordering of the vertices do not matter here, but rather the direction of rotation matters here. In order to get an angle which is $\le 90^{\circ}$ in magnitude, we make the rotation clockwise which results in the value of the angle being negative.

Now in the figure as shown below, $A$ , $B$ and $C$ are ordered in clockwise order. In this case, the value of $\measuredangle ABC$ is actually positive. (The ordering of the vertices does not matter. Only the direction of rotation does.)

$[asy] /* Converted from GeoGebra by User:Azjps using Evan's magic cleaner https://github.com/vEnhance/dotfiles/blob/main/py-scripts/export-ggb-clean-asy.py */ /* A few re-additions are done using bubu-asy.py. This adds the dps, xmin, linewidth, fontsize and directions. https://github.com/Bubu-Droid/dotfiles/blob/master/bubu-scripts/bubu-asy.py */ pair A = (31.64835,16.28958); pair B = (40.11708,2.37181); pair C = (44.07794,21.53331); import graph; size(8cm); pen dps = linewidth(0.5) + fontsize(13); defaultpen(dps); real xmin = -5, xmax = 5, ymin = -5, ymax = 5; draw(arc(B,4.60985,78.32093,121.31981)--B--cycle, linewidth(0.6) + blue); draw(arc(B,4.60985,-101.67906,-58.68018)--B--cycle, linewidth(0.6) + blue); draw(arc(B,2.56591,-58.68018,78.32093)--B--cycle, linewidth(0.6) + red); draw(arc(B,2.56591,121.31981,258.32093)--B--cycle, linewidth(0.6) + red); draw(A--B, linewidth(0.6)); draw(B--C, linewidth(0.6)); draw((37.77614,-8.95301)--B, linewidth(0.6) + linetype("4 4")); draw(B--(45.63427,-6.69530), linewidth(0.6) + linetype("4 4")); draw(arc(B,4.60985,78.32093,121.31981), linewidth(0.6) + blue,EndArcArrow(6)); draw(arc(B,4.60985,-101.67906,-58.68018), linewidth(0.6) + blue); draw(arc(B,4.27057,-101.67906,-58.68018), linewidth(0.6) + blue); draw(arc(B,2.56591,121.31981,251.28453), linewidth(0.6) + red); dot("$C$", A, NW); dot("$B$", B, dir(180)); dot("$A$", C, N); [/asy]$

To make this even more clear, we take an example from EGMO. This example should make the idea even more vivid.

Now let us get to the original question due to which this blog-post was made. How do we define $\measuredangle (\ell_1, \ell_2)$ ? If you've read through the blog-post till here, you should try this little exercise yourself first before moving on.

For doing this, we define $T=\ell_1\cap \ell_2$ . Now we rotate $\ell_1$ w.r.t. $T$ in an anti-clockwise direction till it gets super-imposed on the line $\ell_2$ . Now if the value of the angle is $\le 90^{\circ}$ , that's basically our desired directed angle and the sign also remains positive (since we rotated in an anti-clockwise direction). However, if the value of the angle turns out to be $\ge 90^{\circ}$ , then we take it modulo $180^{\circ}$ and make its magnitude $\le 90^{\circ}$ . In this case, the sign of the angle is negative. Thus this case is equivalent to taking the line $\ell_1$ and rotating it about $T$ in a clockwise direction till it gets super-imposed on $\ell_2$ .

$[asy] /* Converted from GeoGebra by User:Azjps using Evan's magic cleaner https://github.com/vEnhance/dotfiles/blob/main/py-scripts/export-ggb-clean-asy.py */ /* A few re-additions are done using bubu-asy.py. This adds the dps, xmin, linewidth, fontsize and directions. https://github.com/Bubu-Droid/dotfiles/blob/master/bubu-scripts/bubu-asy.py */ import graph; size(8cm); pen dps = linewidth(0.5) + fontsize(13); defaultpen(dps); real xmin = -5, xmax = 5, ymin = -5, ymax = 5; draw(arc((40.11708,2.37181),5.21970,78.32093,121.31981)--(40.11708,2.37181)--cycle, linewidth(0.6) + blue); draw(arc((62.92721,4.40749),5.21970,78.32093,121.31981)--(62.92721,4.40749)--cycle, linewidth(0.6) + red); draw((54.45847,18.32526)--(62.92721,4.40749), linewidth(0.6)); label("$\ell_1$", (54.45847,18.32526)--(62.92721,4.40749), SW); draw((62.92721,4.40749)--(66.88806,23.56899), linewidth(0.6)); label("$\ell_2$", (62.92721,4.40749)--(66.88806,23.56899), SE); draw((60.58626,-6.91732)--(62.92721,4.40749), linewidth(0.6)); draw((62.92721,4.40749)--(68.44439,-4.65961), linewidth(0.6)); draw((31.64835,16.28958)--(40.11708,2.37181), linewidth(0.6)); label("$\ell_2$", (31.64835,16.28958)--(40.11708,2.37181), SW); draw((40.11708,2.37181)--(44.07794,21.53331), linewidth(0.6)); label("$\ell_1$", (40.11708,2.37181)--(44.07794,21.53331), SE); draw((37.77614,-8.95301)--(40.11708,2.37181), linewidth(0.6)); draw((40.11708,2.37181)--(45.63427,-6.69530), linewidth(0.6)); draw(arc((40.11708,2.37181),5.21970,78.32093,119.21372), linewidth(0.6) + blue,EndArcArrow(6)); draw(arc((62.92721,4.40749),5.21970,78.32093,119.21372), linewidth(0.6) + red,BeginArcArrow(6)); [/asy]$

Thus we can conclude our findings for the directed angles as, $\measuredangle (\ell_1,\ell_2)=-\measuredangle (\ell_2,\ell_1)$ and
$\measuredangle (AB,BC)=\measuredangle ABC=-\measuredangle CBA=-\measuredangle (CB,BA) .$
However I should mention that $\measuredangle (AB,BC)$ is the same as $\measuredangle (AB,CB)$ since both the lines $BC$ and $CB$ are basically equivalent.

I hope I was able to convey my message through this blog-post (plix halp this nub to larn bemter englis :maybe:

).

Thank You!!

This post has been edited 8 times. Last edited by kamatadu, Nov 2, 2024, 7:51 PM

math

geometry

angles

directed angles

reminisccence

by HoRI_DA_GRe8, Sep 21, 2024, 10:31 AM

POST BY ANTAREEP

Finally created a math tag on this blog.

So for no particular reason I was thinking today about the past.About what we have done over the years.Then I looked forward 3 months to January Mains and Inmo.4 months to my school life coming to a close.I barely do math these days, dont know if I 'll make INMO this year or not.Heck I dont know if I can even solve oly like before. I have been only watching reels since Inmo results came out.
OK WAIT? Thats not what an ideal student does right? yes they dont.I did , I am stupid and that is well known.Kamatadu will probably make it this year. He is working hard , I really hope he does.

This guy first texted me in 2021 , we were both continuous ioqm failures at that time, and we were as well . UNTILL 2023. UNTILL 2024. This guy made some dumb mistakes in INMO and I still doubt if he didnt would I have been able to make it or not.

Believe me its been a lot . A lot. to be honest math doesnt even interests me these days. Although i still enjoy doing it nevertheless.We have a lot of memories , mtrp 23,24,rsm,sharygin and STEMS 24 .STEMS, oh it was the best and I am definitely not going to dive into the details because it has formed the best part of our memories or atleast I hope so.Now with a few days of oly math remaining , its time to jump in , with everyone of the lot.Jishnu, Oishik,Shameek, Avigyan and even Soham Bhadra. Time for one last race , one last fight..................

I hope everyone is in it and everyone fights as well.

This post has been edited 1 time. Last edited by kamatadu, Nov 2, 2024, 5:20 PM

math

Submit

i feel like i see a shoutout
by Nuterrow, Feb 27, 2025, 4:51 PM
Thanks for the tips
by math_holmes15, Feb 26, 2025, 6:13 AM
Good Luck with your new series
by SomeonecoolLovesMaths, Feb 24, 2025, 11:38 AM
hawsoworzz plij gibb teeps on how to attains skeels
by L13832, Nov 25, 2024, 6:53 AM
E bhai hothat cs change korli ki byapar
by HoRI_DA_GRe8, Nov 3, 2024, 8:03 PM
Ngl that pfp is orz
by factcheck315, Nov 2, 2024, 9:09 PM
finally shouting out orz ppl in orz blog
by Kappa_Beta_725, Oct 25, 2024, 6:13 PM
Omega lavil
Pro lavil
God lavil
Lavil lavil

by Sreemani76, Apr 19, 2024, 7:20 PM
Nice
by Om245, Jan 12, 2024, 2:18 AM
@below We lost finals, so GG...
by kamatadu, Nov 26, 2023, 2:43 PM
We won semis
by Project_Donkey_into_M4, Nov 15, 2023, 6:05 PM
yess sss
by polynomialian, Nov 3, 2023, 3:10 PM
We had a great camp in ISI back in February for MTRP (MATHEMATICS TALENT REWARD PROGRAM). So since the fest season of CMI and isi is near,this means the season is back again!!!.So should we post the mtrp 2023 experience?

Comment!
by HoRI_DA_GRe8, Nov 2, 2023, 4:27 PM
Saar good luck for RMO
haw prep going btw
Me dead non geo :skull:
by BVKRB-, Oct 20, 2023, 2:06 PM
Ors spams to sala tui ekai kortis be

Anyway ors everyone
by HoRI_DA_GRe8, Aug 15, 2023, 3:09 PM
ooo you also trying sharygin poblams nais saar, tell progress lol
by BVKRB-, Jul 5, 2022, 5:36 AM
@orsmax_viewer_sar, me nubmax is trying the sharygin poblams so me numbax not get time to update blog for few days :")
by kamatadu, Jul 3, 2022, 5:20 PM
@below no sar, I waste too much time sar :")
by kamatadu, Jul 1, 2022, 3:25 PM
hello sar bhul gaye kya
by anurag27826, Jun 28, 2022, 2:19 PM
AT LAST!!! Post on SET THEORY is finally complete
by kamatadu, Jun 21, 2022, 3:15 PM
IM BACK! Check the latest post for the description of why I went missing
by kamatadu, Jun 15, 2022, 6:09 PM
Guys im really sorry for not updating the blog for so long, actually ive been busy with fixing my mobile since the past few days, so I couldnt update my blog, I believe I will be done with this work by tomorrow and Ill again start updating the blog
by kamatadu, Jun 13, 2022, 11:24 AM
No bengali plx me no understand single word and me too lazy to use gugul translat
by BVKRB-, Jun 7, 2022, 8:55 AM
Amar dadu bhola dadu bhalo dadu noye.
Kamatadu orz
by Bratin_Dasgupta, May 29, 2022, 6:19 AM
hello sar
by anurag27826, May 26, 2022, 1:09 PM
ughh, my time management is literally the worst, cant even figure out time for updating blog
by kamatadu, May 25, 2022, 6:22 PM
ok saar enuf orsing
by BVKRB-, Mar 4, 2022, 5:39 PM
hello sar, plis domt depress by showing ur pure ors skills here pls
by kamatadu, Feb 27, 2022, 3:58 PM
"Hello World?"
by BVKRB-, Feb 27, 2022, 9:57 AM

50 shouts

amar_04 haw so orsmax :omighty:

Angles in Geometry done correctly

by kamatadu, Nov 2, 2024, 7:35 PM

reminisccence

by HoRI_DA_GRe8, Sep 21, 2024, 10:31 AM