11.3 curves, polygons and symmetrybtravers.weebly.com/.../6729909/curves_polygons_and... · a...
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11.3 Curves, Polygons and Symmetry
Polygons
Simple
DefinitionA shape is simple if it doesn’t cross itself, except maybe at theendpoints.
Closed
DefinitionA shape is closed if the endpoints meet.
Polygon
DefinitionA polygon is a simple closed curve where all sides are line segments.There are no arcs allowed.
Concave v. Convex
DefinitionA polygon is convex if any segment connecting any two points in theinterior of the curve lies entirely inside the curve.
DefinitionA polygon is concave if any segment connecting two points of theinterior of a curve passes outside of the curve
Classification of Polygons by Sides
Number of Sides Polygon Name3
triangle4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagon11 hendecagon12 dodecagonn n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle
4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagon11 hendecagon12 dodecagonn n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle4
quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagon11 hendecagon12 dodecagonn n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle4 quadrilateral5
pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagon11 hendecagon12 dodecagonn n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle4 quadrilateral5 pentagon6
hexagon7 heptagon8 octagon9 nonagon10 decagon11 hendecagon12 dodecagonn n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle4 quadrilateral5 pentagon6 hexagon7
heptagon8 octagon9 nonagon10 decagon11 hendecagon12 dodecagonn n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle4 quadrilateral5 pentagon6 hexagon7 heptagon8
octagon9 nonagon10 decagon11 hendecagon12 dodecagonn n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9
nonagon10 decagon11 hendecagon12 dodecagonn n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10
decagon11 hendecagon12 dodecagonn n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagon11
hendecagon12 dodecagonn n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagon11 hendecagon12
dodecagonn n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagon11 hendecagon12 dodecagonn
n-gon
Classification of Polygons by Sides
Number of Sides Polygon Name3 triangle4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagon11 hendecagon12 dodecagonn n-gon
Regular Polygons
DefinitionA regular polygon is a polygon where all sides and interior angles arecongruent.
What is the name of a regular triangle?
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• •What is the name of a regular quadrilateral?
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• •
Regular Polygons
DefinitionA regular polygon is a polygon where all sides and interior angles arecongruent.
What is the name of a regular triangle?
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• •What is the name of a regular quadrilateral?
• •
• •
Regular Polygons
DefinitionA regular polygon is a polygon where all sides and interior angles arecongruent.
What is the name of a regular triangle?
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• •What is the name of a regular quadrilateral?
• •
• •
Exterior Angle Sum
What is the exterior angle sum for a square?
What is the exterior angle sum for a pentagon?
Conjecture?
Exterior Angle Sum
The exterior angle sum for a convex polygon is 360◦
Exterior Angle Sum
What is the exterior angle sum for a square?
What is the exterior angle sum for a pentagon?
Conjecture?
Exterior Angle Sum
The exterior angle sum for a convex polygon is 360◦
Exterior Angle Sum
What is the exterior angle sum for a square?
What is the exterior angle sum for a pentagon?
Conjecture?
Exterior Angle Sum
The exterior angle sum for a convex polygon is 360◦
Exterior Angle Sum
What is the exterior angle sum for a square?
What is the exterior angle sum for a pentagon?
Conjecture?
Exterior Angle Sum
The exterior angle sum for a convex polygon is 360◦
Triangle Classifications
We can classify triangles in two ways - by the number of congruentsides and by the types of angles. What types of triangles are there?
Types of TrianglesBy Side By Angle
scalene no congruent sides acute all angles acuteisosceles at least two congruent sides right one right angle
equilateral all sides congruent obtuse one obtuse angle
Important to note: your book correctly states that a triangle that isequilateral is also isosceles. Some books incorrectly say that these aredisjoint descriptions.
Triangle Classifications
We can classify triangles in two ways - by the number of congruentsides and by the types of angles. What types of triangles are there?
Types of TrianglesBy Side By Angle
scalene no congruent sides acute all angles acuteisosceles at least two congruent sides right one right angle
equilateral all sides congruent obtuse one obtuse angle
Important to note: your book correctly states that a triangle that isequilateral is also isosceles. Some books incorrectly say that these aredisjoint descriptions.
Triangle Classifications
We can classify triangles in two ways - by the number of congruentsides and by the types of angles. What types of triangles are there?
Types of TrianglesBy Side By Angle
scalene no congruent sides acute all angles acuteisosceles at least two congruent sides right one right angle
equilateral all sides congruent obtuse one obtuse angle
Important to note: your book correctly states that a triangle that isequilateral is also isosceles. Some books incorrectly say that these aredisjoint descriptions.
Quadrilateral Classifications
That we want to do here is list all properties we know about thedifferent quadrilaterals. Included in these properties we want to pointout are:
if opposite sides are parallel
if opposite sides are congruent
if opposite angles are congruent
if adjacent angles are congruent
if adjacent sides are perpendicular
if diagonals bisect each other
if diagonals are perpendicular
if diagonals are congruent
if there is at least one pair of parallel sides
if there is at least one pair of congruent sides
Parallelogram
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
Parallelogram
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
Rectangle
opposite sides congruentopposite angles are congruentdiagonals bisect each otheropposite sides are paralleldiagonals are congruentadjacent sides are perpendicular
Rectangle
opposite sides congruentopposite angles are congruentdiagonals bisect each otheropposite sides are parallel
diagonals are congruentadjacent sides are perpendicular
Rectangle
opposite sides congruentopposite angles are congruentdiagonals bisect each otheropposite sides are paralleldiagonals are congruentadjacent sides are perpendicular
Rhombus
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
adjacent sides congruent
Rhombus
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
adjacent sides congruent
Rhombus
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
adjacent sides congruent
Square
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
diagonals are congruent
adjacent sides are perpendicular
adjacent sides congruent
Square
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
diagonals are congruent
adjacent sides are perpendicular
adjacent sides congruent
Square
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
diagonals are congruent
adjacent sides are perpendicular
adjacent sides congruent
Kite
two pairs of congruent adjacent sides
diagonals are perpendicular
one pair of congruent angles
Kite
two pairs of congruent adjacent sides
diagonals are perpendicular
one pair of congruent angles
Trapezoid
at least one pair of parallel sides
Trapezoid
at least one pair of parallel sides
Isosceles Trapezoid
at least one pair of parallel sides
at least one pair of congruent sidesbase angles congruent
Isosceles Trapezoid
at least one pair of parallel sides
at least one pair of congruent sidesbase angles congruent
Isosceles Trapezoid
at least one pair of parallel sides
at least one pair of congruent sidesbase angles congruent
Hierarchy for Triangles
For triangles, here is the relationship between them based on sides.
Triangle
Scalene Triangle Isosceles Triangle
Equilateral Triangle
Hierarchy for Quadrilaterals
Can you come up with a similar diagram for quadrilaterals?
Quadrilateral
Kite Trapezoid
Parallelogram Isosceles Trapezoid
Rhombus Rectangle
Square
Hierarchy for Quadrilaterals
Can you come up with a similar diagram for quadrilaterals?
Quadrilateral
Kite Trapezoid
Parallelogram Isosceles Trapezoid
Rhombus Rectangle
Square
Symmetries
1 Line symmetry2 Rotational symmetry3 Point symmetry
Line Symmetry
Line SymmetryA figure has line symmetry if we can draw a line that divides thefigure in half. We can think of this as the line over which we can foldthe figure to make it fold onto itself.
How Many Lines of Symmetry?
How Many Lines of Symmetry?
Rotational Symmetry
Rotational (Turn) SymmetryA figure has rotational symmetry if we can rotate the figure somenumber of degrees less than 360◦ and get the same exact polygonback in the same orientation.
We describe the property as α degrees of turn symmetry.
Rotational Symmetry?
Rotational Symmetry?
Point Symmetry
Point Symmetry
Point symmetry is the term that we use when a figure has 180◦ turnsymmetry.
Which of our figures have point symmetry?
Diagonals
DiagonalA diagonal is a line segment that joins to nonadjacenet vertices.
The question is, how many diagonals does a convex polygon have?
Number of Diagonals
A convex polygon with n sides we have n(n−3)2 diagonals.
Diagonals
DiagonalA diagonal is a line segment that joins to nonadjacenet vertices.
The question is, how many diagonals does a convex polygon have?
Number of Diagonals
A convex polygon with n sides we have n(n−3)2 diagonals.
Diagonals
DiagonalA diagonal is a line segment that joins to nonadjacenet vertices.
The question is, how many diagonals does a convex polygon have?
Number of Diagonals
A convex polygon with n sides we have n(n−3)2 diagonals.