Given a regular 90-gon, we construct 90 isosceles triangles on the exterior of the polygon such that each isosceles triangle has an edge of the polygon as its base and has legs formed by the extensions of the two adjacent sides. Measure of the largest angle of one such triangle is ______ degrees.

Each interior angle of a regular 90-gon is $$180^\circ - \frac{360^\circ}{90} = 176^\circ$$. At each end of the chosen edge the extension of the adjacent side makes an angle of $$180^\circ - 176^\circ = 4^\circ$$ with that edge, so both base angles are $$4^\circ$$. The apex angle is therefore $$180^\circ - 8^\circ = 172^\circ$$, which is the largest angle.