## Thinking Mathematically (6th Edition)

(a) A polygon is a two dimensional figure which is of two types that is regular polygon and irregular polygon. Regular polygon is a figure in which all the sides are of same length. In irregular polygon all sides are of different length. A polygon with three sides is called a triangle.A polygon with four sides is called rectangle or quadrilateral.A polygon with five sides is called pentagon and so on.A tessellations is a type of art that is used to define a relationship between geometry and the visual arts. Tessellations are created by repeated use of same figures that will leave no gap and no overlaps and thus cover whole plane. To create a tessellation the primary requirement is that the sum of the measures of the angles of a regular polygon that are together at each vertex must be$360{}^\circ$. Hence, the name of the regular polygon that surround each vertex is Hexagons and Triangles. (b) The number of angles that come together at each vertex are five in which four angles are formed by the triangle, one is formed by the hexagon. Measure of an angle of a regular Hexagon will be determined by dividing the sum of the measures of all angles which is $720{}^\circ$ by its sides i.e. 6. \begin{align} & \text{m}\measuredangle \text{A=}\frac{\text{72}{{\text{0}}^{\text{o}}}}{\text{6}} \\ & \text{=12}{{\text{0}}^{\text{o}}} \end{align} Measure of an angle of a regular triangle will be determined by dividing the sum of the measures of all angles which is $180{}^\circ$ by its sides i.e. 3. \begin{align} & m\measuredangle A=\frac{180{}^\circ }{3} \\ & =60{}^\circ \end{align} There are four angles that are formed by the triangle. Hence, the measure of the angles at each vertex of a polygon is $120{}^\circ$, $60{}^\circ$, $60{}^\circ$,$60{}^\circ$,$60{}^\circ$. (c) A tessellations is a type of art that is used to define a relationship between geometry and the visual arts. Tessellations are created by repeated use of same figures that will leave no gap and no overlaps and thus cover whole plane. To create a tessellation the primary requirement is that the sum of the measures of the angles of a regular polygon that are together at each vertex must be$360{}^\circ$. To check whether a tessellation can be created or not add the sum of the angles. \begin{align} & \text{Sum of the angles}=120{}^\circ +60{}^\circ +60{}^\circ +60{}^\circ +60{}^\circ \\ & =360{}^\circ \end{align} Hence, the creation of a tessellation is possible.