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## Isosceles Problems

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 $\angle CDA$?  That is, what is the value of $x$?

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 $\overline{AD}$ and $\overline{BC}$ as $K$, then $\angle CKD$ and $\angle AKB$ are both $70^{\circ}$, $\angle CKA$ and $\angle DKB$ are both $110^{\circ}$, which makes $\angle KCA$ $50^{\circ}$ and $\angle ADB$ is $40^{\circ}$.

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…