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6.1 Reflection-Symmetric
Figures
Chapter 6 Polygons and
Symmetry
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Example of numbers with line
symmetry:
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2 Cases
A
A’
or
B
C
C’
6.1 Reflection-Symmetric
Figures
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D=D’
B’
A=A’
C=C’
B=B’
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What about a circle?
6.1 Reflection-Symmetric
Figures
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Trace each capitol letter below, page
301, and draw all symmetry lines.
A C X
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These examples have line symmetry. The
figure can be reflected over a line so that
one of these 2 cases occur.
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6.1 Reflection-Symmetric
Figures
Segment Symmetry Theorem
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6.1 Reflection-Symmetric
Figures
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Every segment has exactly 2 symmetry
lines:
1.
2.
Circle Symmetry Theorem
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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
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Angle Symmetry Theorem
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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
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Symmetric Figures Theorem
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If a figure is symmetric, then any pair of
corresponding parts under the symmetry
are congruent.
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A
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What does this mean?
Given: line AE is a symmetry
line for ? ABC
C
What can you prove?
A
B
E
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6.1 Reflection-Symmetric
Figures
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What can you prove?
6.2 Isosceles Triangles
AB @AC
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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.
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A
C
B
E
6.2 Isosceles Triangles
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6.2 Isosceles Triangles
Def. Isosceles Triangle
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Isosceles Triangle Coincidence Theorem
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us
di
ra
s
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diu
ra
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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.
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6.2 Isosceles Triangles
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Median
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The segment connecting a vertex of the
triangle to the midpoint of the opposite
side.
6.2 Isosceles Triangles
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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
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Isosceles Triangle Base Angles Theorem
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if a triangle has two congruent sides, then
the angle’
s opposite them are congruent.
Triangle-Sum Theorem
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The sum of the measures of the angles of
a triangle is 180º.
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Example 1:
6.2 Isosceles Triangles
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But ? XYZ is
isosceles (XY = ZY)
with vertex angel Y.
So, from the
Isosceles Triangle
Base Theorem, m<X
= m<Z.
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Thus,
mÐ X + mÐ X = 157
2mÐ X = 157
mÐ X = 78.5
X
Y
Z
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6.2 Isosceles Triangles
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Equilateral Triangle Symmetry Theorem
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Every equilateral triangle has three symmetry lines,
which are the bisectors of its angles (or equivalently,
the perpendicular bisectors of its sides).
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Equilateral Triangle Angle Theorem
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Corollary-(a theorem that follows immediately from another
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6.3 Types of Quadrilaterals
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Rectangle
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A quadrilateral is a
rectangle if and
only if it has four
right angles.
Square
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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.)
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Each angle of an equilateral triangle has measure 60º.
6.3 Types of Quadrilaterals
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Parallelogram
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A quadrilateral is a
parallelogram if
and only if both pairs
of its opposite sides
are parallel.
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Rhombus
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A quadrilateral is a
rhombus if and only
if its four sides are
equal in length.
6.3 Types of Quadrilaterals
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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
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6.3 Types of Quadrilaterals
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Venn Diagram
6.3 Types of Quadrilaterals
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Trapezoid
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rhombus
squares
rectangles
6.3 Types of Quadrilaterals
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Kite
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A quadrilateral is a
kite if and only if it
has two distinct pairs
of consecutive sides
of the same length.
Convex
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nonconvex
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Isosceles trapezoid
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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
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Quadrilateral Hierarchy Theorem
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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.
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6.3 Types of Quadrilaterals
6.4 Properties of Kites
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Kite Symmetry Theorem
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Symmetry diagonal
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Kite Diagonal Theorem
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6.4 Properties of Kites
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Four radii
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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?
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6.4 Properties of Kites
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6.5 Properties of Trapezoids
E
A
Rhombusn
Each diagonal of a rhombus is the perpendicular
bisector of the other diagonal.
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2
1
B
D
C
Proof of a Theorem
Given: AB DC , AD has been extended to point E.
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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.
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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-
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Quadrilateral with at least one pair of
parallel sides.
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Trapezoid Angle Theorem
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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.
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Makes trapezoid properties even more valuable.
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6.5 Properties of Trapezoids
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Isosceles Trapezoid Symmetry Theorem
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6.5 Properties of Trapezoids
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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
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The perpendicular bisectors of the sides of
a rectangle are symmetry lines for the
rectangle.
A
D
6.5 Properties of Trapezoids
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Isosceles Trapezoid Theorem
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B
C
6.7 Regular Polygons
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In an isosceles trapezoid, the non-base
sides are congruent.
Regular Polygons
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a convex polygon whose angles are all congruent
and whose sides are all congruent.
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Equilateral
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Equiangular
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If all sides of the polygon have the same length.
If all angles of the polygon have the same
measure.
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6.7 Regular Polygons
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Regular n-gons
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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
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Center of a Regular Polygon Theorem
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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
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(n –
2) 180 where n = number of sides
Regular Polygon Symmetry Theorem
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–
every regular n-gon possesses
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(3)180 = 540º
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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
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