Transcript File

Polygons
1
Polygons
Definition: A closed figure formed by line segments so that each
segment intersects exactly two others, but only at their
endpoints.
These figures are not polygons
These figures are polygons
2
Classifications of a Polygon
Convex: No line containing a side of the polygon contains a point
in its interior
Concave:
A polygon for which there is a line
containing a side of the polygon and
a point in the interior of the polygon.
3
Classifications of a Polygon
Regular: A convex polygon in which all interior angles have the
same measure and all sides are the same length
Irregular: Two sides (or two interior angles) are not congruent.
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Polygon Names
3 sides
Triangle
4 sides
Quadrilateral
5 sides
Pentagon
6 sides
Hexagon
7 sides
Heptagon
8 sides
Octagon
9 sides
Nonagon
10 sides
Decagon
12 sides
n sides
Dodecagon
n-gon
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Regular Polygons
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Regular polygons have:
• All side lengths congruent
• All angles congruent
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Area of Regular Polygon

Apothem of a polygon: the distance from
the center to any side of the polygon.
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Area of Regular Polygon

We can now subdivide the polygon into
triangles.
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Area  s  a  n
2
s  side _ length
a  apothem
n  Number _ of _ sides
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Triangles and Quadrilaterals
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Classifying Triangles by Sides
Scalene: A triangle in which all 3 sides are different lengths.
A
A
B
C
BC = 3.55 cm
B
C
BC = 5.16 cm
Isosceles: A triangle in which at least 2 sides are equal.
G
Equilateral: A triangle in which all 3 sides are equal.
GH = 3.70 cm
H
HI = 3.70 cm
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I
Classifying Triangles by Angles
Acute: A triangle in which all 3 angles are less than 90˚.
G
76
57
47
H
Obtuse:
I
A
A triangle in which one and only one
angle is greater than 90˚& less than 180˚
44
28 108 C
B
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Classifying Triangles by Angles
Right: A triangle in which one and only one angle is 90˚
A
56
B
90
34
C
Equiangular: A triangle in which all 3 angles are the same measure.
B
60
A
60
60
C
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Classification by Sides
with Flow Charts & Venn Diagrams
polygons
Polygon
triangles
Triangle
scalene
Scalene
Isosceles
isosceles
equilateral
Equilateral
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Classification by Angles
with Flow Charts & Venn Diagrams
Polygon
polygons
triangles
Triangle
right
acute
Right
Obtuse
Acute
Equiangular
equiangular
obtuse
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What is a Quadrilateral?
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All quadrilaterals have four
sides.
They also have four angles.
The sum of the four angles
totals 360°
These properties are what
make quadrilaterals alike,
but what makes them
different?
Parallelogram

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Two sets of parallel sides
Two sets of congruent sides.
The angles that are opposite
each other are congruent
(equal measure).
The two theorems below can also be used to show that
a given quadrilateral is a parallelogram.
Rectangle

Has all properties of quadrilateral and
parallelogram

A rectangle also has four right angles.

A rectangle can be referred to as an
equiangular parallelogram because all
four of it’s angle are right, meaning they
are all 90° (four equal angles).
6-4 Properties of Special Parallelograms
Since a rectangle is a parallelogram by Theorem 6-4-1,
a rectangle “inherits” all the properties of
parallelograms that you learned in Lesson 6-2.
Holt Geometry
6-4 Properties of Special Parallelograms
Example 1: Craft Application
A woodworker constructs a
rectangular picture frame so
that JK = 50 cm and JL = 86
cm. Find HM.
Rect.  diags. 
KM = JL = 86
Def. of  segs.
 diags. bisect each other
Substitute and simplify.
Holt Geometry
Rhombus
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A rhombus is sometimes referred to as a
“slanted square”.
A rhombus has all the properties of a
quadrilateral and all the properties of a
parallelogram, in addition to other properties.
A rhombus is often referred to as a
equilateral parallelogram, because it has four
sides that are congruent (each side length has
equal measure).
Like a rectangle, a rhombus is a parallelogram. So you
can apply the properties of parallelograms to
rhombuses.
Example 2A: Using Properties of Rhombuses to Find
Measures
TVWX is a rhombus.
Find TV.
WV = XT
13b – 9 = 3b + 4
10b = 13
b = 1.3
Def. of rhombus
Substitute given values.
Subtract 3b from both sides and
add 9 to both sides.
Divide both sides by 10.
Example 2A Continued
TV = XT
Def. of rhombus
TV = 3b + 4
Substitute 3b + 4 for XT.
TV = 3(1.3) + 4 = 7.9 Substitute 1.3 for b and simplify.
Square
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The square is the most specific member of
the family of quadrilaterals. The square
has the largest number of properties.

Squares have all the properties of a
quadrilateral, all the properties of a
parallelogram, all the properties of a
rectangle, and all the properties of a
rhombus.

A square can be called a rectangle,
rhombus, or a parallelogram because it
has all of the properties specific to those
figures.
Trapezoid
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Unlike a parallelogram,
rectangle, rhombus, and
square who all have two sets of
parallel sides, a trapezoid only
has one set of parallel sides.
These parallel sides are
opposite one another. The
other set of sides are non
parallel.
Isosceles Trapezoid
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One can never assume a trapezoid is
isosceles unless they are given that the
trapezoid has specific properties of an
isosceles trapezoid.
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Isosceles is defined as having two equal
sides. Therefore, an isosceles trapezoid has
two equal sides. These equal sides are
called the legs of the trapezoid, which are
the non-parallel sides of the trapezoid.
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Both pair of base angles in an isosceles
trapezoid are also congruent.
Right Trapezoid
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A right trapezoid also has one set of
parallel sides, and one set of nonparallel sides.
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A right trapezoid has exactly two right
angles. This means that two angles
measure 90°.
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There should be no problem identifying
this quadrilateral correctly, because it’s
just like it’s name. When you think of
right trapezoid, think of right angles!
Quadrilateral Family Tree
Quadrilateral
Parallelogram
Rectangle
Square
It’s important to have a good
understanding of how each of the
quadrilaterals relate to one another.
Trapezoid
Any quadrilateral that has two sets of
parallel sides can be considered a
parallelogram.
A rectangle and rhombus are both types
of parallelograms, and a square can be
considered a rectangle, rhombus, and a
parallelogram.
Rhombus
Isosceles
Right
Trapezoid
Trapezoid
Any quadrilateral that has one set of
parallel sides is a trapezoid. Isosceles and
Right are two types of trapezoids.
Class Quiz
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A walkway 3.0 m wide is constructed along
the outside edge of a square courtyard. If the
perimeter of the courtyard is 320 m, what is
the perimeter of the square formed by the
outer edge of the walkway?