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?

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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?

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• •

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.