An arrangement which can be readily seen to be valid albeit not minimal may be formed by observing that joining two "small" (3-matchstick) triangles and two "large" (9-matchstick) rigid triangles will yield a construct with four vertices of order two (the remaining vertices are all of order four). The order-two vertices will form a trapezoid whose non-parallel edges are of equal length and form an angle which may, by flexing the joints, may be adjusted within a range that includes 45 degrees and extends almost up to 60. One may thus fasten seven or eight such constructs together to yield a "donut" which meets the necessary conditions.
The figure on the left has the angles adjusted so as to show that the range extends slightly beyond 45 degrees. Other constructs may use fewer line segments, but this construct is probably the "simplest".
Another useful building block is this combination of four large and two small triangles:
While this section, unlike the one above, isn't directly flexible, it only has two order-two nodes rather than four. As a consequence, three of them may be combined to yield a regular graph, or--more interestingly--four of them may be combined to yield a flexible regular graph.