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6.1 Reflection-Symmetric Figures Chapter 6 Polygons and Symmetry n Example of numbers with line symmetry: 0 n 2 Cases A A’ or B C C’ 6.1 Reflection-Symmetric Figures n D=D’ B’ A=A’ C=C’ B=B’ 0 What about a circle? 6.1 Reflection-Symmetric Figures n 0 Trace each capitol letter below, page 301, and draw all symmetry lines. A C X n N These examples have line symmetry. The figure can be reflected over a line so that one of these 2 cases occur. 1 6.1 Reflection-Symmetric Figures Segment Symmetry Theorem n n 6.1 Reflection-Symmetric Figures n Every segment has exactly 2 symmetry lines: 1. 2. Circle Symmetry Theorem n Its perpendicular bisector The line containing the segment Just a few of the lines through the center of the circle. 6.1 Reflection-Symmetric Figures n Angle Symmetry Theorem n The line containing the bisector of an angle is a symmetry line of the angle. A circle has infinitely many lines of symmetry, all through the center of the circle. 6.1 Reflection-Symmetric Figures n Symmetric Figures Theorem n If a figure is symmetric, then any pair of corresponding parts under the symmetry are congruent. n n A n What does this mean? Given: line AE is a symmetry line for ? ABC C What can you prove? A B E 2 6.1 Reflection-Symmetric Figures n What can you prove? 6.2 Isosceles Triangles AB @AC n n Ð C @Ð B Cannot say CE @EB because these are not parts of the triangle. Isosceles Symmetry Theorem The line containing the bisector of the vertex angle of an isosceles triangle is a symmetry line for the triangle. n A C B E 6.2 Isosceles Triangles n 6.2 Isosceles Triangles Def. Isosceles Triangle n n Isosceles Triangle Coincidence Theorem n us di ra s n n diu ra n 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. 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 6.2 Isosceles Triangles n Median n The segment connecting a vertex of the triangle to the midpoint of the opposite side. 6.2 Isosceles Triangles n Z In circle Y, m<Y = 23. Y What is m<X? n By the Triangle-Sum Theorem: mÐ Y + mÐ X + mÐ Z = 180 n Substitution Property 12 + mÐ X + mÐ Z = 180 mÐ X + mÐ Z = 157 n Addition property of equality 6.2 Isosceles Triangles n Isosceles Triangle Base Angles Theorem n n if a triangle has two congruent sides, then the angle’ s opposite them are congruent. Triangle-Sum Theorem n The sum of the measures of the angles of a triangle is 180º. X Example 1: 6.2 Isosceles Triangles n But ? XYZ is isosceles (XY = ZY) with vertex angel Y. So, from the Isosceles Triangle Base Theorem, m<X = m<Z. n Thus, mÐ X + mÐ X = 157 2mÐ X = 157 mÐ X = 78.5 X Y Z 4 6.2 Isosceles Triangles n Equilateral Triangle Symmetry Theorem n Every equilateral triangle has three symmetry lines, which are the bisectors of its angles (or equivalently, the perpendicular bisectors of its sides). n Equilateral Triangle Angle Theorem n Corollary-(a theorem that follows immediately from another n 6.3 Types of Quadrilaterals n Rectangle n n A quadrilateral is a rectangle if and only if it has four right angles. Square n A quadrilateral is a square if and only if it has four equal side and four right angles. If a triangle is equilateral, then it is equiangular. theorem.) n Each angle of an equilateral triangle has measure 60º. 6.3 Types of Quadrilaterals n Parallelogram n A quadrilateral is a parallelogram if and only if both pairs of its opposite sides are parallel. n Rhombus n A quadrilateral is a rhombus if and only if its four sides are equal in length. 6.3 Types of Quadrilaterals n 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 6.3 Types of Quadrilaterals n Venn Diagram 6.3 Types of Quadrilaterals n Trapezoid n rhombus squares rectangles 6.3 Types of Quadrilaterals n Kite n n A quadrilateral is a kite if and only if it has two distinct pairs of consecutive sides of the same length. Convex n nonconvex n Isosceles trapezoid n A trapezoid is an isosceles trapezoid if and only if it has a pair of base angles equal in measure. 6.3 Types of Quadrilaterals n Quadrilateral Hierarchy Theorem n n A quadrilateral is a trapezoid if and only if it has at least one pair of parallel sides. The seven types of quadrilaterals are related as shown in the hierarchy picture which will follow. Page 319 every rhombus is a special kite. 6 6.3 Types of Quadrilaterals 6.4 Properties of Kites n Kite Symmetry Theorem n n Symmetry diagonal n Kite Diagonal Theorem n n 6.4 Properties of Kites n Four radii n The line containing the ends of a kite is a symmetry line for the kite. The diagonal determined by the ends. 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 Triangle Reflection Ends –the common endpoint of the equal sides of a kite. How many “ ends” does a rhombus have? 7 6.4 Properties of Kites n 6.5 Properties of Trapezoids E A Rhombusn Each diagonal of a rhombus is the perpendicular bisector of the other diagonal. n 2 1 B D C Proof of a Theorem Given: AB DC , AD has been extended to point E. n n Conclusion m<1 = m<2 = 180º m<2 = m<D <1 and <D are supplementary This leads us to the Trapezoid Angle Thm. 1. 2. 3. 4. n n 1. 2. 3. 4. Justification Supplementary <‘ s lines ¦ ? CA’ s Substitution Definition Supplementary <‘ s 6.5 Properties of Trapezoids 6.5 Properties of Trapezoids Trapezoids- n n Quadrilateral with at least one pair of parallel sides. n Trapezoid Angle Theorem n In a trapezoid consecutive angles between a pair of parallel sides are supplementary. Any property of all trapezoids holds for all parallelograms, rhombuses, rectangles, squares, and isosceles trapezoids. n Makes trapezoid properties even more valuable. 8 6.5 Properties of Trapezoids n Isosceles Trapezoid Symmetry Theorem n 6.5 Properties of Trapezoids n The perpendicular bisector of one base of an isosceles trapezoid is the perpendicular bisector of the other base and symmetry line for the trapezoid. Rectangle Symmetry Theorem n The perpendicular bisectors of the sides of a rectangle are symmetry lines for the rectangle. A D 6.5 Properties of Trapezoids n Isosceles Trapezoid Theorem n B C 6.7 Regular Polygons n In an isosceles trapezoid, the non-base sides are congruent. Regular Polygons n a convex polygon whose angles are all congruent and whose sides are all congruent. n Equilateral n Equiangular n n If all sides of the polygon have the same length. If all angles of the polygon have the same measure. 9 6.7 Regular Polygons n Regular n-gons n n n n n n n n n n n n n n n n 6.7 Regular Polygons = 3 equilateral triangle =4 square = 5 regular hexagon = 6 regular hexagon = 7 regular heptagon = 8 regular octagon = 9 regular nonagon = 10 regular dodecagon 6.7 Regular Polygons n Center of a Regular Polygon Theorem n 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 n n (n – 2) 180 where n = number of sides Regular Polygon Symmetry Theorem n – every regular n-gon possesses n (3)180 = 540º n 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