To solve this, we must understand some basic geometry concepts related to circles and triangles.
1. First of all, because the segments AB and AC are related to congruent arcs AB and AC, we can imply that they are chords in a circle.
2. According to the properties of a circle, if two chords are of the same length (in this case AB and AC, given that their corresponding arcs are congruent), they subtend equal angles at the center of the circle. This leads to the formation of isosceles triangles when we join the center of the circle to the endpoints of the chords.
3. In our current situation, the circle is circumscribed around the triangle ABC, and the given arcs AB and AC are of equal measure.
4. Arc AB corresponds to the angle subtended at the center of the circle O by the line segment AB. Similarly, arc AC corresponds to the angle subtended by the line segment AC at the center of the circle O.
5. Given that arc AB is congruent to arc AC, they subtend equal angles at the center O. But these equivalent angles at the circle's center are also congruent to their corresponding angles within the triangle ABC at vertices B and C, respectively. (This is because in the same or equal circles, equal arcs subtend equal angles at the center.)
6. Hence, the angles at vertices B and C in triangle ABC are equal, which establishes that triangle ABC is an isosceles triangle.
So, Noah knows that the top layer of his cake, which is triangle ABC, is an isosceles triangle.