Download Polygon

Properties
and Angle
Attributes
of Polygons
6.1 Polygon
Sum Theorems
I can classify polygons based on their sides and angles.
Do Now:
1. A
?
is a three-sided polygon.
2. A
?
is a four-sided polygon.
Success Criteria:
 I can identify sum of interior
angles
 I can identify polygons by sides
and angles
triangle
quadrilateral
Today’s Agenda
 New seats
 Do now
 Lesson
 HW #
6.1 Polygon
Sum Theorems
Properties
andAngle
Attributes
of Polygons
I can classify polygons based on their sides and angles.
Do Now:
1. Name the polygon by the number of its
sides. Then tell whether the polygon is
regular or irregular, concave or convex. nonagon; irregular;
concave
2. Find the sum of the interior angle
measures of a convex 11-gon.
1620°
Success Criteria:
 I can identify sum of interior
angles
 I can identify polygons by sides
and angles
Today’s Agenda
 Do Now
 Check HW #42
 Test corrections
 No homework
Polygon
– A closed plane
formed by
or more
Properties
and figure
Attributes
of 3Polygons
segments
Each segment that forms a polygon is a side of the
polygon.
The common endpoint of two sides is a vertex of the
polygon.
A segment that connects any two nonconsecutive
vertices is a diagonal.
Properties
and
You can
name a polygon
by the number of its
sides. The table shows
the names of some
common polygons.
Attributes of Polygons
Remember!
A polygon is a closed plane figure formed by
three or more segments that intersect only at
their endpoints.
Example 1A: Identifying Polygons
Properties and Attributes of Polygons
Discuss with A/B partner whether the figure is
a polygon. If it is a polygon, name it by the
number of sides.
polygon, hexagon
not a polygon
polygon, heptagon
not a polygon
not a polygon
Properties and Attributes of Polygons
All the sides are congruent in an equilateral polygon.
All the angles are congruent in an equiangular polygon.
A regular polygon is one that is both equilateral and
equiangular. If a polygon is not regular, it is called
irregular.
Properties and Attributes of Polygons
A polygon is concave if any part of a diagonal
contains points in the exterior of the polygon. If no
diagonal contains points in the exterior, then the
polygon is convex. A regular polygon is always
convex.
Checkand
It Out!
Example 2aof Polygons
Properties
Attributes
Discuss with A/B partner whether the polygon is
regular or irregular and whether it is concave or
convex.
regular, convex
irregular, concave
Properties and Attributes of Polygons
To find the sum of the interior angle measures of a
convex polygon, draw all possible diagonals from
one vertex of the polygon. This creates a set of
triangles. The sum of the angle measures of all the
triangles equals the sum of the angle measures of
the polygon.
Properties and Attributes of Polygons
In each convex polygon, the number of triangles
formed is two less than the number of sides n. So
the sum of the angle measures of all these triangles
is (n — 2)180°.
Properties and Attributes of Polygons
Example 3A: Finding Interior Angle Measures and
Sums in Polygons
Find the sum of the interior angle measures of a
convex heptagon (7-sides).
(n – 2)180°
Polygon  Sum Thm.
(7 – 2)180°
A heptagon has 7 sides,
so substitute 7 for n.
900°
Simplify.
Properties and Attributes of Polygons
Example 3B: Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each interior angle of a
regular 16-gon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
Polygon  Sum Thm.
(16 – 2)180° = 2520°
Substitute 16 for n
and simplify.
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 16.
Properties and Attributes of Polygons
Example 3C: Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each
interior angle of pentagon
ABCDE.
(5 – 2)180° = 540° Polygon  Sum Thm.
Polygon 
mA + mB + mC + mD + mE = 540° Sum Thm.
35c + 18c + 32c + 32c + 18c = 540
135c = 540
c=4
Substitute.
Combine like terms.
Divide both sides by 135.
Properties and Attributes of Polygons
Example 3C Continued
mA = 35(4°) = 140°
mB = mE = 18(4°) = 72°
mC = mD = 32(4°) = 128°
Properties and Attributes of Polygons
In the polygons below, an exterior angle has been
measured at each vertex. Notice that in each case,
the sum of the exterior angle measures is 360°.
Properties and Attributes of Polygons
Remember!
An exterior angle is formed by one side of a
polygon and the extension of a consecutive side.
Properties and Attributes of Polygons
Example 4A: Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each exterior angle of a
regular 20-gon.
A 20-gon has 20 sides and 20 vertices.
sum of ext. s = 360°.
measure of one ext.  =
Polygon  Sum Thm.
A regular 20-gon
has 20  ext. s, so
divide the sum by
20.
The measure of each exterior angle of a regular
20-gon is 18°.
Properties and Attributes of Polygons
Example 5: Art Application
Ann is making paper stars for
party decorations. What is the
measure of 1?
1 is an exterior angle of a regular
pentagon. By the Polygon Exterior
Angle Sum Theorem, the sum of the
exterior angles measures is 360°.
A regular pentagon has 5 
ext. , so divide the sum by 5.
Properties and Attributes of Polygons
Assignment #2
pg 356 #7 - 25
Properties and Attributes of Polygons
Example 4B: Finding Interior Angle Measures and
Sums in Polygons
Find the value of b in polygon
FGHJKL.
Polygon Ext.  Sum Thm.
15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360°
120b = 360
b=3
Combine like terms.
Divide both sides by 120.
Properties
Attributes
of Polygons
Exteriorand
Angles
Activity
Get graph paper or
Cut out each exterior
scratch paper.
angle
Using a straight edge
Put the colored angles
draw a polygon, each
together
person in your group
must have different
amount of sides.
What happened?
Discuss who will draw
each shape. It does not What about the other
polygons?
have to be regular
What can you conclude
After the shape is
about the exterior angle
drawn extend each
consecutive side.
of polygons? 3 sides? 4
sides? 5 sides?
Highlight the exterior
angle
Properties and Attributes of Polygons
Exterior Angles Activity cont.
Cut out each exterior angle
Put the colored angles together
What happened?
What about the other polygons?
What can you conclude about the exterior
angle of polygons? 3 sides? 4 sides? 5 sides?
Properties and Attributes of Polygons
 Take out your Polygon Exploration
worksheet.
 With your group yesterday what
observations can you make about
the degrees of a polygon?
 The person with their birthday
closest to today will write your
groups conclusion or findings from
your Polygon Exploration.
Properties and Attributes of Polygons
Do Now
1. Name the polygon by the number
of its sides. Then tell whether the
polygon is regular or irregular,
concave or convex.
nonagon; irregular; concave
2. Find the sum of the interior angle
measures of a convex 11-gon. 1620°
3. Find the measure of each interior angle of a
regular 18-gon. 160°
4. Find the measure of each exterior angle of a
regular 15-gon. 24°
Properties and Attributes of Polygons