During the long easter weekend, I finally have time to check out the web and forums and I found this nice little geometry problem on triangles. At first glance the problem looks difficult, however it is not too bad after figuring out some of the angles and realizing couple isosceles triangles in it.

**Solution**

First of all let’s label the points A, B, C, D, E, and F in the diagram.

Obviously, triangle ABC, triangle BCD, and triangle ABF are isosceles triangles. Let

and let

In isosceles triangle ABC, we have

In isosceles triangle BCD, we have

Then apply the Product-to-Sum identity

We have

Now apply the cosine law on triangle ACD,

And then finally apply sine law in triangle ABD,

Then apply double angle identity,

Therefore angle *x* is 30 degrees.

**Remarks:**

Some additional beautiful results can be found if we solve *x* differently, but I will leave this for next time. 🙂

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*Related*

Useful info. Hope to see more good posts in the future.

Notice that this doesn’t work though. You have base angles of 36 and 48 on an ISOSCELES triangle. This doesn’t make sense because base angles are equal. Thus, the given information cannot possibly be true.

NVM.