Saturday, December 2, 2017

Another Geometry Problem (Extension) : Part-2

I was pleasantly surprised to know about the amount of interest / attention drawn by my previous post on the geometric problem on various facebook groups... So many people had not just read and liked it, but had even left their comments with their approach of solving this problem. I would first like to thank all these people for sharing their methods. 

I did share some of the different methods with my students (construction of auxiliary lines in different ways) and they were quite intrigued by this fact that there can be so many ways of seeing and solving a geometric problem. 

For those who haven't read the previous post or are not aware of the original problem, you can find it here:

However, Special Thanks to following remark made by a professor/ researcher Michael de Villiers on this post:

"Good, but did you perhaps encourage your students to generalize further by exploring more 'kinks' as in this online, dynamic activity?"

This motivated me to try the extension of this problem with my students...

So then the next day, 

I asked them -- 

"The other day, in our original problem (fig (1)), when there were just two inclined lines between the two parallel lines, we saw that the measure of angle b was sum of the measures of angles a and c......, What will happen if there are three such lines between these parallel lines as in fig (2)...... or may be if there are four or more such lines between them as in fig 3 and fig 4? ........ I mean, will there still be any relation between these angles a, b, c and d in fig (2), the way we had  a + c = b in fig (1) ?"

They thought for a while and started working on their notebooks.. I stopped them...

"Wait. Can you first guess what could be the possible relation?"

These were their different guesses about the four angles in fig (2)

i) a + b = c + d

ii) a + d = b + c 

iii) a + b + c + d = 180

Then I asked them to continue with their work...

While some were engrossed in arriving at the solution, I could see that few were unable to find a break though... So I asked them to "actually construct this diagram using a scale and then measure the angles a,b, c and d to really find out if there is any relation..."

This idea was well received by this bunch... However I could see that not all were ready to do this construction / verification exercise....

So I could see the class divided into two groups - One who were working on the geometric solution without any construction, and the Other group who was doing the construction to 'actually see' the relation first....

Almost both the groups arrived at their respective results simultaneously i.e.

sum of the measures of angles a and c equals the sum of the measures of angles b & d.
a + c = b + d

And then each of these groups was asked to try with the other method - for verification of their result arrived using the first method. It was a beautiful exercise...

Finally they presented their approach to the class :

Kanchan's way

You can see that she has extended the two inclined lines to meet the parallel lines to form two triangles. Then she claimed that the angles 180-(180-c+d) and 180-(180-b+a) will be equal as they form a pair of alternate angles and hence, d-c = a-b  means 

a + c = b + d


While Vaishnavi had construction the auxiliary lines in a different way:

Vaishnavi's way
She has extended the upper inclined line beyond the upper horizontal line and then dropped a perpendicular from there to the lower horizontal line.

She now saw a pentagon and used the property of interior angles of pentagon to prove that a + c = b + d


Jitu instead drew two lines which were parallel to the two given parallel lines as follows: 

Jeetu's way
He then defined a new angle 'x' as interior alternate angles between these two newly constructed parallel lines.

Now if you focus on the upper two parallel lines, then x = b - a  while if you look at the two lower parallel lines, x = c - d , thus leading to 

b - a = c - d   which means a + c = b + d


It didn't take much time for them to arrive at the relation between angles in fig (3) and fig (4) i.e. more kinks between the two parallel lines. This is what one of them did immediately: 

She proved that 

a + c + e = b + d

and even others could infer the relation for more number of angles in fig (4),  

a + c + e = b + d + f


I would also like to share that before I gave these new set of (extended) problems to them, I also showed them couple of different approaches of solving the original problem that they had already solved the other day:

And if you observe closely, this will reveal to you the secret behind Vaishnavi's idea/ approach to solve the extended problems.

In fact, Its also interesting to note that the three methods of drawing the auxiliary lines used by the students in solving this extended problem are completely different than those used by them in solving the original problem (fig (1) few days back.


While we all could happily derive the relation between the angles that will be formed due to many kinks , I wanted my students to 'wonder' at this result..... I asked them if this result /relation was obvious / natural or does it 'surprise' them or 'makes them wonder' as to why this must happen so?

And this nasty question of mine made them stare at this problem again :-) :-) :-)


Meanwhile, what are your thoughts/ reactions about this result? Does that make you 'wonder'?   :-)


Rupesh Gesota

1 comment:

  1. Cool extension!

    After reading their proofs, my first thought is: the kinky angles seem to me like measures of tilty-ness.

    We start with a line. Then we tilt it one way, then another, back and forth. In the end, we come back parallel to what we started with.

    So intuitively, it makes sense that the tilts in one direction should match the tilts in the other direction, to cancel each other out.

    But I didn't think of this until after I saw the proofs. So even though it seems intuitive in hindsight, it wasn't obvious to me at first. Isn't that often the way with math?