[5] (a) A nonagon divides up (as in Figure 2.8, p. 64) into 7 triangles, so the 9 vertex angles all together sum to 7*180 = 1260 degrees. Thus in a regular nonagon, each vertex angle measures 1260/9 = 140 degrees.

(b) If 3078 were the measure of the sum of all n vertex angles, then we would have (n-2)180 = 3078, so n = (3078/180) + 2 = 19.1. But this is too small since we know 3078 is one angle short. So let's try n = 20 to see if it works. If not, then we'll try n = 21, n=22, etc. until we find the right one. If n = 20, then the sum of the 20 vertex angles is (n-2)180 = 18*180 = 3240, so each vertex angle in a regular 20-gon measures 3240/20 = 162, and if we take all but one we get (20 - 1)162 = 19*162 = 3078 which is right. So n = 20 is correct.

[10] Using the formula (n - 2)180/n for the vertex angle of a regular n-gon, we find that the vertex angle for a regular hexagon is 120 degrees and for a regular octagon it is 135 degrees. From the drawing we are given we see that x + 120 + 135 = 360 so x = 105, and we see that y + 120 + 120 = 360 so y = 120. The shaded region is thus not a regular pentagon, since the angles are not all the same (one is 105 degrees and another is 120 degrees).

[13] (a) The exterior angle in a regular hexagon is 180 - 120 = 60 degrees.

(b) The exterior angle is 180 - 144 = 36, so, using the formula given right before the statement of this problem (on p. 80), we see the number of sides is n = 360/36 = 10.

(c) Using the formula again, we get n = 360/40 = 9.

Section 2.2

[12] (a) This strip pattern has translational symmetry only so the notation is p111.

(b) This one has translational and rotational, so we get p112.

(c) This one has translational and a glide reflection only, so we get p1a1.

(d) This one has translational, vertical reflection, horizontal reflection and rotational symmetries, so we get pmm2.

[14] (a) Of the letters shown, O, T, U, V, W, X, Y have vertical reflection symmetry.

(b) Only O and X have horizontal reflection symmetry.

(c) Only N, O, S, X and Z have rotational symmetry.

[16] (a) The only lines of reflection symmetry are the diagonals through opposite corners.

(b) The only rotational symmetry is the 180 degree rotation through the center of the square.

[18] (a) The perpendicular bisector of any side of any of the hexagons is a line of reflection symmetry, and so is the angle bisector of any vertex angle of any of the hexagons.

(b) Any rotation of 60, 120, 180, 240 or 300 degrees about the center of any hexagon is a rotational symmetry. Also rotations of 120 or 240 degrees about the center of any little triangle is a rotational symmetry. The only other rotational symmetry is a 180 degree rotation about any vertex of any little triangle.