6.1 Reflection-Symmetric Figures. Chapter 6 Polygons and Symmetry. 6.1 Reflection-Symmetric Figures. 6.1 Reflection-Symmetric Figures

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1 Chapter 6 Polygons and Symmetry Example of numbers with line symmetry: What about a circle? 2 Cases D=D Trace each capitol letter below, page 301, and draw all symmetry lines. B C C B or = B=B C=C C X N These examples have line symmetry. The figure can be reflected over a line so that one of these 2 cases occur. 1

2 Segment Symmetry Theorem Every segment has exactly 2 symmetry lines: 1. Its perpendicular bisector 2. The line containing the segment Circle Symmetry Theorem circle has infinitely many lines of symmetry, all through the center of the circle. Just a few of the lines through the center of the circle. ngle Symmetry Theorem The line containing the bisector of an angle is a symmetry line of the angle. Symmetric Theorem If a figure is symmetric, then any pair of corresponding parts under the symmetry are congruent. What does this mean? Given: line E is a symmetry line for? BC What can you prove? C E B 2

3 What can you prove? C Isosceles Symmetry Theorem Cannot say EB because these are not Ð B The line containing the bisector of the vertex angle of an isosceles triangle is a symmetry line for the triangle. parts of the triangle. C E B Def. Isosceles Triangle at least two sides equal vertex angle- the angle determined by the equal sides in an isosceles triangle. base- the side opposite the vertex. base angels the two angles whose vertices are the endpoints of the base. radius radius Isosceles Triangle Coincidence Theorem In an isosceles triangle, the bisector of the vertex angle, the perpendicular bisector of the base, and the median to the base determine the same line. 3

4 Median The segment connecting a vertex of the triangle to the midpoint of the opposite side. Example 1: In circle Y, m<y = 23. What is m<x? By the Triangle-Sum Theorem: Substitution Property ddition property of equality Y mð Y + mð X + mð Z = mð X + mð Z = 180 X mð X + mð Z = 157 Z Isosceles Triangle Base ngles Theorem if a triangle has two congruent sides, then the angle s opposite them are congruent. Triangle-Sum Theorem The sum of the measures of the angles of a triangle is 180º. But? XYZ is isosceles (XY = ZY) with vertex angel Y. So, from the Isosceles Triangle Base Theorem, m<x = m<z. Thus, mð X + mð X = 157 X 2mÐ X = 157 mð X = 78.5 Y Z 4

5 Equilateral Triangle Symmetry Theorem Every equilateral triangle has three symmetry lines, which are the bisectors of its angles (or equivalently, the perpendicular bisectors of its sides). Equilateral Triangle ngle Theorem If a triangle is equilateral, then it is equiangular. Corollary-(a theorem that follows immediately from another theorem.) Each angle of an equilateral triangle has measure 60º. Rectangle rectangle if and only if it has four right angles. Square square if and only if it has four equal side and four right angles. Parallelogram parallelogram if and only if both pairs of its opposite sides are parallel. Rhombus rhombus if and only if its four sides are equal in length. If a figure is of any type in the hierarchy, it is also a figure of all types connected above it in the hierarchy. rhombus rectangle square 5

6 Venn Diagram rhombus squares rectangles Trapezoid trapezoid if and only if it has at least one pair of parallel sides. Isosceles trapezoid trapezoid is an isosceles trapezoid if and only if it has a pair of base angles equal in measure. Kite kite if and only if it has two distinct pairs of consecutive sides of the same length. Convex nonconvex every rhombus is a special kite. Quadrilateral Hierarchy Theorem The seven types of quadrilaterals are related as shown in the hierarchy picture which will follow. Page 319 6

7 6.4 Properties of Kites Kite Symmetry Theorem The line containing the ends of a kite is a symmetry line for the kite. Symmetry diagonal The diagonal determined by the ends. Kite Diagonal Theorem The symmetry diagonal of a kite is the perpendicular bisector of the other diagonal and bisects the two angle s at the ends of the kite. 6.4 Properties of Kites 6.4 Properties of Kites Four radii Triangle Reflection Ends the common endpoint of the equal sides of a kite. How many ends does a rhombus have? 7

8 Rhombus- 6.4 Properties of Kites Each diagonal of a rhombus is the perpendicular bisector of the other diagonal. 6.5 Properties of Trapezoids E 2 1 Proof of a Theorem D C Given: B DC, D has been extended to point E. B Conclusion 1. m<1 = m<2 = 180º 2. m<2 = m<d 3. <1 and <D are supplementary 4. This leads us to the Trapezoid ngle Thm. Justification 1. Supplementary < s 2. lines? C s 3. Substitution 4. Definition Supplementary < s 6.5 Properties of Trapezoids Trapezoids- Quadrilateral with at least one pair of parallel sides. ny property of all trapezoids holds for all parallelograms, rhombuses, rectangles, squares, and isosceles trapezoids. Makes trapezoid properties even more valuable. 6.5 Properties of Trapezoids Trapezoid ngle Theorem In a trapezoid consecutive angles between a pair of parallel sides are supplementary. 8

9 6.5 Properties of Trapezoids Isosceles Trapezoid Symmetry Theorem The perpendicular bisector of one base of an isosceles trapezoid is the perpendicular bisector of the other base and symmetry line for the trapezoid. 6.5 Properties of Trapezoids Rectangle Symmetry Theorem The perpendicular bisectors of the sides of a rectangle are symmetry lines for the rectangle. B D C 6.5 Properties of Trapezoids Isosceles Trapezoid Theorem In an isosceles trapezoid, the non-base sides are congruent. 6.7 Regular Polygons Regular Polygons a convex polygon whose angles are all congruent and whose sides are all congruent. Equilateral If all sides of the polygon have the same length. Equiangular If all angles of the polygon have the same measure. 9

10 6.7 Regular Polygons Regular n-gons n = 3 equilateral triangle n =4 square n = 5 regular hexagon n = 6 regular hexagon n = 7 regular heptagon n = 8 regular octagon n = 9 regular nonagon n = 10 regular dodecagon 6.7 Regular Polygons Center of a Regular Polygon Theorem In any regular polygon there is a point (its center) which is equidistant from all of its vertices. Draw a regular pentagon. 6.7 Regular Polygons Polygon Sum Theorem (n 2) 180 where n = number of sides (3)180 = 540º 6.7 Regular Polygons Regular Polygon Symmetry Theorem every regular n-gon possesses 1. n symmetry lines, which are the perpendicular bisectors of each of its sides and the bisectors of each of its angles; 2. n-fold rotation symmetry. (2) 180 =360º 10