## Isosceles Problems

January 6, 2017 by richbeveridge

One of my students gave me a problem last fall that was very interesting.

The problem is posed with the following diagram:

What is the measure of ? That is, what is the value of ?

I’ve given a similar problem in my trigonometry class for the past few years, except that version of the problem has a side length included and the triangle is not isosceles.

A pdf of this problem is linked below:

jun_7_mth_112_river_problem

Working from my experience with the other version of this problem, I began to write in values for the various unlabeled angles in the diagram – if we label the intersection of and as , then and are both , and are both , which makes and is .

I added in new variables and created a system of four equations with four unknowns, but it was a dependent system.

The solution for this problem that was devised by the student who gave it to me is after the jump…

His solution was primarily synthetic and not algebraic!

His first step was to draw line segment such that and are congruent, creating a small isosceles triangle in the interior of the larger one. Then and are both and is

Once this is done, we can add back in and look at triangle . Notice that and , so that is also , making isosceles with sides and congruent.

Now, if we draw , we know that , and are all congruent. With and congruent, then is isosceles. With vertex angle , then both and are as well, making equilateral with side congruent to , and .

Then, we add back into the diagram, is now as the original angle was and is .

So, with redrawn and , then in triangle , and , then must also be , making isosceles and sides and congruent.

But side is already congruent to in the equilateral triangle , so that means that and are congruent and is isosceles as well. Once we know this, the solution is trivial – with , then must be split equally between angles and , so and !

The reasoning here is somewhat tricky and convoluted, but I looked for a simple algebraic solution to this for hours and got nowhere.

In researching this problem I did run across a similar but different problem.

Chris Harrow, Mathematics Chair at the Hawken School in Cleveland, OH, has a math blog called CAS Musings. Here, he ponders a problem involving isosceles triangles with no angle measures given. This problem generates a system of 6 equation with 6 unknowns that he solves using Wolfram Alpha.

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