1. 12/8/2015 3.1: Symmetry in Polygons

2. 12/8/20153.1: Symmetry in Polygons On the first day of school, Mr Vilani gave his 3 rd grade students 5 new words to spell. On each school day after that he gave them 3 new words to spell. In the first 20 days of school, how many new words had the students been given to spell?

3. 12/8/20153.1: Symmetry in Polygons Polygons A figure is a polygon iff it is a plane figure formed from 3 or more segments such that each segment intersects exactly 2 others, one at each endpoint, and no 2 segments with a common endpoint are collinear. The segments are the sides of the polygon and the endpoints are the vertices of the polygon.

4. 12/8/20153.1: Symmetry in Polygons Classifying Polygons Number of Sides Polygon 3Triangle 4Quadrilateral 5Pentagon 6Hexagon 7Heptagon 8Octagon 9Nonagon

5. 12/8/20153.1: Symmetry in Polygons Classifying Polygons Number of Sides Polygon 10Decagon 1111-gon 12Dodecagon 1313-gon nn-gon

6. 12/8/20153.1: Symmetry in Polygons Triangles Defn: A triangle is scalene iff it has no congruent sides. A triangle is isosceles iff it has at least 2 congruent sides. A triangle is equilateral iff all 3 sides are congruent.

7. 12/8/20153.1: Symmetry in Polygons Equilateral Polygons Defn: A polygon is equilateral iff all of its sides are congruent.

8. 12/8/20153.1: Symmetry in Polygons Equiangular Polygons Defn: A polygon is equiangular iff all of its angles are congruent.

9. 12/8/20153.1: Symmetry in Polygons Regular Polygons Defn: A polygon is regular iff it is equilateral and equiangular.

10. 12/8/20153.1: Symmetry in Polygons Center of a Regular Polygon Defn: The center of a regular polygon is the point that is equidistant from all vertices of the polygon. Center of the regular polygon

11. 12/8/20153.1: Symmetry in Polygons Central Angle of a Regular Polygon Defn: A central angle of a regular polygon is an angle whose vertex is the center of the regular polygon and whose sides contain 2 consecutive vertices. Central angle of the regular polygon

12. 12/8/20153.1: Symmetry in Polygons The Central Angle of a Regular Polygon The measure, , of a central angle of a regular polygon with n sides is given by the formula:  =  = 360 n

13. 12/8/20153.1: Symmetry in Polygons What is the measure of the central angle of a regular octagon?

14. 12/8/20153.1: Symmetry in Polygons Reflectional Symmetry Defn: A figure has reflectional symmetry iff its reflected image across a line coincides exactly with the preimage. The line is called an axis of symmetry.

15. 12/8/20153.1: Symmetry in Polygons Triangle Symmetry Conjecture An axis of symmetry in a triangle is the perpendicular bisector of the side it intersects, and it passes through the vertex of the angle opposite that side of the triangle. An equilateral triangle has 3 axes of symmetry. A strictly isosceles triangle has 1 axis of symmetry. A scalene triangle has 0 axes of symmetry.

16. 12/8/20153.1: Symmetry in Polygons Axis of Symmetry for a Triangle axis of symmetry

17. 12/8/20153.1: Symmetry in Polygons By the Triangle Symmetry Property, is the yellow segment an axis of symmetry for the triangle? Why or why not?

18. 12/8/20153.1: Symmetry in Polygons By the Triangle Symmetry Property, is the yellow segment an axis of symmetry for the triangle? Why or why not?

19. 12/8/20153.1: Symmetry in Polygons Rotational Symmetry Defn: A figure has rotational symmetry iff it has at least one rotation image not counting rotation images of 0° or multiples of 360° that coincide with the preimage.

20. 12/8/20153.1: Symmetry in Polygons Rotational Symmetry

21. 12/8/20153.1: Symmetry in Polygons Rotational Symmetry All geometric figures have 0° (360°) rotational symmetry. If a figure has only 0° (360°) rotational symmetry, it is said to have trivial rotational symmetry.

22. 12/8/20153.1: Symmetry in Polygons Rotational Symmetry n-fold rotational symmetry: A figure that is said to have n-fold rotational symmetry will rotate onto itself n times if the figure is rotated 360°.

23. 12/8/20153.1: Symmetry in Polygons Center of Symmetry The center of the rotation which yields the rotational symmetry is also called the center of symmetry.

24. 12/8/20153.1: Symmetry in Polygons Describe all symmetries for a regular hexagon. 6-fold rotational symmetry (60°, 120°, 180°, 240°, 300° and 360°=0°) 6 axes of symmetry: the 3 perpendicular bisectors of each pair of opposite sides and the 3 lines containing the center and a vertex.

25. 12/8/20153.1: Symmetry in Polygons Assignment Pages 143- 146, # 10 – 26 (evens), 28 – 32 (all), 34 – 40 (evens), 46 – 50 (all), 52, 54