Thanks to Georgie and Makenzie who pointed out that:

There are four different types of triangle in the circle.

Thanima from Eastlea Community School went a little further and worked out that:

When the band is changed form one peg to the next the difference in the angle
at the centre will be 40 degrees:

360(the whole circle) / 9(number of pegs)

so the angle at the centre for each of the triangles will be 40, 80,
120 and 160 degrees.

Kirsty from Hertfordshire and Essex High School used this to help her find the angles of the two triangles that appear at the end of the problem:

I worked out that in the triangles at the end of the problem the angles
could be worked out by using the isosceles triangles that we found in
class. The isosceles triangles were found by using the centre peg in the
circle and two of the other pegs. I found four isosceles triangles in the
ten peg circle.

The pink triangle is made up of three isosceles triangles.

One triangle
had the centre angle of 80 degrees making the other two angles 50 degrees.

The other isosceles triangle in the pink triangle has a centre angle of 120
degrees making the other two angles 30 degrees.

The final isosceles
triangle in the pink triangle has a centre angle of 160 degrees making the
other angles 10 degrees.

This makes the 3 angles of the pink triangle 60 degrees (50 degrees plus 10 degrees), 40 degrees (10 degrees plus 30 degrees) and 80 degrees (30 degrees plus 50 degrees).

The blue triangle is made up of two isosceles triangles minus one isosceles triangle.

One triangle has a centre angle of 40 degrees making
the other two angles 70 degrees.

The other isosceles triangle has a centre
angle of 80 degrees making the other angles 50 degrees.

The isosceles
triangle that has the angles that have to be taken away to find the angles of the blue
triangle has a centre angle of 120 degrees making the other two angles 30
degrees.

This makes the 3 angles of the blue triangle 120 degrees (70 degrees plus 50 degrees), 40 degrees (70 degrees minus 30 degrees) and 20 degrees (50 degrees minus 30 degrees).