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Angles in Polygons
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Find the measures of interior and exterior angtes of polygons.
THEOREM 8.1: POLyGON TNTER|OR ANGLES THEOREM
words The sum of the measures of the
angles of a convex' polygon
rrser rJ
rY''Dv' with ra sides
Symbols
| '
mlL + ml2 + ... +
interior
is
'r
(,/-.\
a\
V1:,
mln :
Follow-Up
The Triangle sum Theorem states that the sum of the measures
o.
of the angles of a triangle
To use Theorem g.1 with a
triangle, substitute
for n.
is
_
(n -'2). 18Oo: (- - 2).18Oo:
.18O' : _o /
The Quadrilateral lnterior Angles Theorem states that the sum of
o.
the measures of the interior angles of a quadrilateral is
To use Theolem 8.1 with a quadrilateral, substitute tor n.
(n-2).18O':(
-2).
.
Use
lgoo
:
-'
18o" =
'/
tfte Polygon lnterior AngtesTheorem
Find the sum of the measures of the interior angles
of the convex heptagon shown.
Solution
A heptagon has _ sides. use the polygon lnteriof Angles Theorem
and substitute tor n.
Answer The sum of the measures of the interior angles of the
o.
convex heptagon
is
Lesson
8.2
Geometry, Concepts and Skills Notetaking
Guide
lg3
Follow-Up
ln Example 1,.is your answer true lor Any convex heptagon?
Find the Measure of an lnfcrtor An$le
lA
Find the measure of
in the diagram.
142"
Solution
1360
_
sides. Use the Polygon
The polygon has
for n.
lnterior Angles Theorem and substitute
(n -,2).180o: (
136'
_
- 2).18Oo:-'180o:
Use your answer to write an equation for the sum of the measures
mlA.
mlA +' mlB + mlC + mlD + mlE + mlF :
o:
o+
o+
o+
o+
mlA +
of the interior angles. Solve lor
o
o
mlA +
o
C.
ot.a Regular
lnteriot
Find the measure of an interior angle of a
regular octafon.
Solution
An octagon has
;
sides. Use the Poly$on
lnterior Angles Theorem and substitute , for n.
{n
- 2)'
18O"
:
(
. 2)'180o :
'
18Oo
-
Because the octa$on is regular, each angle has the same measure.
So, divide to find the measure of one interior angle.
Answer The measure of an interior angle of a re$ular octagon
!-
184
I
Geometry, Concepts and
O
Sklls Notetaklng Guide Chapter 8
.
Follow-llp Complete the table for a regular n-gon.
n
3
4
5
6
7
8
9
Sum ofthe
measures of
the interior
angles
Measure of
one interioi
angle
O
Cne,ckpoint.ln Exercises 1-3, find the meas ute ol
lG.
4. Find the measure of an interior angle of a regutar polygon with
twelve sides.
8.2: POIYGON EXTERTOR ANctES THEOREM
Wbrds The sum of the measures of the exterior
angles of a convex polygon, one angle at each
THEOREIUI
is
Symbols mlt + ml2 + ... + mln :
vertex,
o.
Lesson
8.2 .
GeometrS Goncefis and Skllls l{otetaklng
Gutde
1g5
Find the value of x.
Solution
Using.the Polygon Exterior An$les Theorem, set the sum of the
o.
measures of the exterior hn$les equal to
95' +
85o
+
2xo
* xo :
o
PolygonEtteriorAnglesfheorem
Combine like terms.
from each slde.
Dldde each side B
Ansrver The value of
_.
.
x is
Fottow-llp Check your answer for Example 4.
95o+85'+ f, +X:95o+85''+
O
cnec@oint Find the vilue of x.
Geometry; Conoepts ahd Skllls Notetaking
I
r
Gulde
Chapter 8
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