Download 11.3 Curves, Polygons and Symmetry

11.3 Curves, Polygons and Symmetry
Polygons
Simple
Definition
A shape is simple if it doesn’t cross itself, except maybe at the
endpoints.
Closed
Definition
A shape is closed if the endpoints meet.
Polygon
Definition
A polygon is a simple closed curve where all sides are line segments.
There are no arcs allowed.
Concave v. Convex
Definition
A polygon is convex if any segment connecting any two points in the
interior of the curve lies entirely inside the curve.
Definition
A polygon is concave if any segment connecting two points of the
interior of a curve passes outside of the curve
Classification of Polygons by Sides
Number of Sides
3
Polygon Name
Classification of Polygons by Sides
Number of Sides
3
Polygon Name
triangle
Classification of Polygons by Sides
Number of Sides
3
4
Polygon Name
triangle
Classification of Polygons by Sides
Number of Sides
3
4
5
Polygon Name
triangle
quadrilateral
Classification of Polygons by Sides
Number of Sides
3
4
5
6
Polygon Name
triangle
quadrilateral
pentagon
Classification of Polygons by Sides
Number of Sides
3
4
5
6
7
Polygon Name
triangle
quadrilateral
pentagon
hexagon
Classification of Polygons by Sides
Number of Sides
3
4
5
6
7
8
Polygon Name
triangle
quadrilateral
pentagon
hexagon
heptagon
Classification of Polygons by Sides
Number of Sides
3
4
5
6
7
8
9
Polygon Name
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
Classification of Polygons by Sides
Number of Sides
3
4
5
6
7
8
9
10
Polygon Name
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
Classification of Polygons by Sides
Number of Sides
3
4
5
6
7
8
9
10
11
Polygon Name
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
Classification of Polygons by Sides
Number of Sides
3
4
5
6
7
8
9
10
11
12
Polygon Name
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
hendecagon
Classification of Polygons by Sides
Number of Sides
3
4
5
6
7
8
9
10
11
12
n
Polygon Name
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
hendecagon
dodecagon
Classification of Polygons by Sides
Number of Sides
3
4
5
6
7
8
9
10
11
12
n
Polygon Name
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
hendecagon
dodecagon
n-gon
Regular Polygons
Definition
A regular polygon is a polygon where all sides and interior angles are
congruent.
What is the name of a regular triangle?
Regular Polygons
Definition
A regular polygon is a polygon where all sides and interior angles are
congruent.
What is the name of a regular triangle?
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•
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What is the name of a regular quadrilateral?
Regular Polygons
Definition
A regular polygon is a polygon where all sides and interior angles are
congruent.
What is the name of a regular triangle?
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•
•
What is the name of a regular quadrilateral?
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Exterior Angle Sum
What is the exterior angle sum for a square?
Exterior Angle Sum
What is the exterior angle sum for a square?
What is the exterior angle sum for a pentagon?
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
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 congruent
sides and by the types of angles. What types of triangles are there?
Triangle Classifications
We can classify triangles in two ways - by the number of congruent
sides and by the types of angles. What types of triangles are there?
Types of Triangles
scalene
isosceles
equilateral
By Side
no congruent sides
at least two congruent sides
all sides congruent
acute
right
obtuse
By Angle
all angles acute
one right angle
one obtuse angle
Triangle Classifications
We can classify triangles in two ways - by the number of congruent
sides and by the types of angles. What types of triangles are there?
Types of Triangles
scalene
isosceles
equilateral
By Side
no congruent sides
at least two congruent sides
all sides congruent
acute
right
obtuse
By Angle
all angles acute
one right angle
one obtuse angle
Important to note: your book correctly states that a triangle that is
equilateral is also isosceles. Some books incorrectly say that these are
disjoint descriptions.
Quadrilateral Classifications
That we want to do here is list all properties we know about the
different quadrilaterals. Included in these properties we want to point
out 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
Parallelogram
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
Rectangle
Rectangle
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
Rectangle
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
diagonals are congruent
adjacent sides are perpendicular
Rhombus
Rhombus
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
Rhombus
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
adjacent sides congruent
Square
Square
opposite sides congruent
opposite angles are congruent
diagonals bisect each other
opposite sides are parallel
diagonals are congruent
adjacent sides are perpendicular
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
Kite
two pairs of congruent adjacent sides
diagonals are perpendicular
one pair of congruent angles
Trapezoid
Trapezoid
at least one pair of parallel sides
Isosceles Trapezoid
Isosceles Trapezoid
at least one pair of parallel sides
Isosceles Trapezoid
at least one pair of parallel sides
at least one pair of congruent sides
base 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?
Hierarchy for Quadrilaterals
Can you come up with a similar diagram for quadrilaterals?
Quadrilateral
Kite
Trapezoid
Parallelogram
Rhombus
Isosceles Trapezoid
Rectangle
Square
Symmetries
1
Line symmetry
2
Rotational symmetry
3
Point symmetry
Line Symmetry
Line Symmetry
A figure has line symmetry if we can draw a line that divides the
figure in half. We can think of this as the line over which we can fold
the figure to make it fold onto itself.
How Many Lines of Symmetry?
How Many Lines of Symmetry?
Rotational Symmetry
Rotational (Turn) Symmetry
A figure has rotational symmetry if we can rotate the figure some
number of degrees less than 360◦ and get the same exact polygon
back 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◦ turn
symmetry.
Which of our figures have point symmetry?
Diagonals
Diagonal
A diagonal is a line segment that joins to nonadjacenet vertices.
Diagonals
Diagonal
A diagonal is a line segment that joins to nonadjacenet vertices.
The question is, how many diagonals does a convex polygon have?
Diagonals
Diagonal
A 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.