Question

Isosceles triangle ABC has been dissected into thirteen isosceles acute triangles, as shown in the two following figures. Given that ∠CAB is an integral number of degrees, what is the size of ∠CAB ?

![Isosceles triangle ABC has been dissected into thirteen isosceles acute
triangles, as shown in the two following figures, where all segments of the same length are marked the same way, and the second figure shows the details
of the dissection of △EFG. Given that ∠CAB is an integral number of degrees, what is the size of ∠CAB ?

Answer 1

`/_CAB = 29^@`

Explanation:

Writing `x` for `/_ CAB`, we can annotate the diagram with the other angles like this...

Then we can derive a couple of constraints from the requirement that the isosceles triangles are acute:

From the top diagram, we require `3x < 90^@` and hence:

`x < 30^@`

From the final apex angle in the bottom diagram, we require:

`450^@-25/2x < 90^@`

Hence:

`25/2x > 360^@`

So:

`x > 2/25 * 360^@ = 28.8^@`

So the only possible integer value of `x` is `29^@`