GEOMETRY HELP Identify the repeating figures and a transformation in the tessellation. A repeated combination of an octagon and one adjoining square will completely cover the plane without gaps or overlap. The arrow shows a translation. Quick Check Tessellations LESSON 9-7 Additional Examples

GEOMETRY HELP Determine whether a regular 15-gon tessellates a plane. Explain. Because the figures in a tessellation do not overlap or leave gaps, the sum of the measures of the angles around any vertex must be 360°. Check to see whether the measure of an angle of a regular 15-gon is a factor of 360. Because 156 is not a factor of 360, a regular 15-gon will not tessellate a plane. Use the formula for the measure of an angle of a regular polygon. 180 (n – 2) n a = Substitute 15 for n. 180 (15 – 2) 15 a = Simplify.a = 156 Quick Check Tessellations LESSON 9-7 Additional Examples

GEOMETRY HELP List the symmetries in the tessellation. Starting at any vertex, the tessellation can be mapped onto itself using a 180° rotation, so the tessellation has point symmetry centered at any vertex. The tessellation also has translational symmetry, as can be seen by sliding any triangle onto a copy of itself along any of the lines. Quick Check Tessellations LESSON 9-7 Additional Examples