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Stand Quietly
Lesson 3.3 Angles of
Polygons
Warm-Up #40
(1/13/17)
1. What is the rule for sum of the interior angles of a
triangle?
2. ∑6 + ∑7 + ∑8 =______
3. Label vertex,
edge(side), angle
Warm-Up #42 (1/19/17)
π‘†π‘œπ‘™π‘£π‘’ π‘“π‘œπ‘Ÿ π‘₯ π‘Žπ‘›π‘‘ 𝑓𝑖𝑛𝑑 π‘‘β„Žπ‘’ π‘šπ‘–π‘ π‘ π‘–π‘›π‘” π‘Žπ‘›π‘”π‘™π‘’(𝑠)
Homework (1/19/17)
Lesson 3.3 Packet
Page 1 and 2 ODD
Naming Polygons
closed figure in a plane formed by segments, called sides(edge).
A polygon is a _____________
sides or ______.
angles
A polygon is named by the number of its _____
Naming Polygons
A vertex is the point
of intersection of
two sides.
Consecutive vertices are
the two endpoints of any
side.
Q
P
R
U
A segment whose
endpoints are
nonconsecutive
vertices is a
diagonal.
T
S
Sides that share a vertex
are called consecutive
sides.
sides congruent.
An equilateral polygon has all _____
An equiangular polygon has all angles
______ congruent.
equiangular
equilateral and ___________.
A regular polygon is both ___________
equilateral
but not
equiangular
equiangular
but not
equilateral
regular,
both equilateral
and equiangular
Investigation: As the number of sides of a series of regular polygons increases, what do you
notice about the shape of the polygons?
Naming Polygons
Prefixes are also used to name other polygons.
Prefix
Number of
Sides
Name of
Polygon
tri-
3
triangle
quadri-
4
quadrilateral
penta-
5
pentagon
hexa-
6
hexagon
hepta-
7
heptagon
octa-
8
octagon
nona-
9
nonagon
deca-
10
decagon
Angles of Polygon Investigation.
1.Each partner will have a POSTER and a DIAGRAM.
2. WRITE your NAME on the poster. (Horizontal)
3. You have 10 minutes to COPY the chart, DRAW the
diagram, and WRITE the answers on the poster.
4. PRESENT the poster to the class.
Choose one vertex and draw all possible diagonals from that
vertex
Expectations for Presentation?
Diagonals and Angle Measure
Make a table like the one below.
. 1) Choose one vertex and draw all
possible diagonals from that vertex
2) How many triangles are formed?
Convex
Polygon
Number
of Sides
quadrilateral
4
Number of Diagonals
from One Vertex
1
Number of
Triangles
2
Sum of
Interior Angles
2(180) = 360
Diagonals and Angle Measure
1) Draw a convex pentagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals
from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral
4
1
2
2(180) = 360
pentagon
5
2
3
3(180) = 540
Diagonals and Angle Measure
1) Draw a convex hexagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals
from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral
4
1
2
2(180) = 360
pentagon
5
2
3
3(180) = 540
hexagon
6
3
4
4(180) = 720
Diagonals and Angle Measure
1) Draw a convex heptagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals
from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral
4
1
2
2(180) = 360
pentagon
5
2
3
3(180) = 540
hexagon
6
3
4
4(180) = 720
heptagon
7
4
5
5(180) = 900
Diagonals and Angle Measure
1) Any convex polygon.
2) All possible diagonals from one vertex.
3) How many triangles?
Convex
Polygon
Number
of Sides
quadrilateral
4
1
2
2(180) = 360
pentagon
5
2
3
3(180) = 540
hexagon
6
3
4
4(180) = 720
heptagon
7
4
5
5(180) = 900
n-gon
n
n-3
n-2
(n – 2)180
Theorem
Number of Diagonals
from One Vertex
Number of
Triangles
Sum of
Interior Angles
If a convex polygon has n sides, then the sum of the
measure of its interior angles is (n – 2)180.
Regular Polygons vs Irregular Polygons
https://www.mathsisfun.com/definitions/irregular-polygon.html
Regular or Irregular?
YouTube: interior angles of polygons
https://www.youtube.com/watch?v=G44lR8yR3Vk
https://www.youtube.com/watch?v=m1BXpAnD-1Q
https://www.youtube.com/watch?v=j5jkWFy323U
https://www.youtube.com/watch?v=j5jkWFy323U
Practice problems
http://www.mathworksheets4kids.com/polygon.php
If a convex polygon has n sides, then the
Theorem
sum of the measure of its interior angles is
of interior
(n – 2)180.
angles
Ex. 1: Finding measures of Interior Angles of Polygons
β€’ Find the value of x in the diagram
shown:
142ο‚°
88ο‚°
136ο‚°
105ο‚°
136ο‚°
xο‚°
23
SOLUTION:
β€’ The sum of the
measures of the interior
angles of any hexagon is
(6 – 2) ● 180ο‚° = 4 ● 180ο‚°
= 720ο‚°.
β€’ Add the measure of
each of the interior
angles of the hexagon.
142ο‚°
88ο‚°
136ο‚°
105ο‚°
136ο‚°
xο‚°
24
SOLUTION:
136ο‚° + 136ο‚° + 88ο‚° + 142ο‚° +
105ο‚° +xο‚° = 720ο‚°.
The sum is 720ο‚°
607 + x = 720
Simplify.
X = 113
Subtract 607 from each
side.
β€’The measure of the sixth interior angle of the hexagon is
113ο‚°.
25
EXTERIOR ANGLE THEOREMS
26
Ex. 2: Finding the Number of Sides of a Polygon
β€’ The measure of each interior angle is 140ο‚°. How many sides does the
polygon have?
28
Solution:
( n ο€­ 2)(180)
n
= 140ο‚°
(n – 2) ●180ο‚°= 140ο‚°n
Corollary to Thm. 11.1
Multiply each side by n.
180n – 360 = 140ο‚°n
40n = 360
n=9
Distributive Property
Addition/subtraction props.
Divide each side by 40.
29
Ex. 3: Finding the Measure of an Exterior Angle
30
Ex. 3: Finding the Measure of an Exterior Angle
31
Ex. 3: Finding the Measure of an Exterior Angle
32
Using Angle Measures in Real Life
Ex. 4: Finding Angle measures of a polygon
33
Using Angle Measures in Real Life
Ex. 5: Using Angle Measures of a Regular Polygon
34
Using Angle Measures in Real Life
Ex. 5: Using Angle Measures of a Regular Polygon
35
Using Angle Measures in Real Life
Ex. 5: Using Angle Measures of a Regular Polygon
Sports Equipment: If you were designing the
home plate marker for some new type of ball
game, would it be possible to make a home
plate marker that is a regular polygon with
each interior angle having a measure of:
a. 135°?
b. 145°?
36
Using Angle Measures in Real Life
Ex. : Finding Angle measures of a polygon
37
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