geometry section 6-1 1112
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Angles of PolygonTRANSCRIPT
Chapter 6Quadrilaterals
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Tuesday, April 10, 2012
Section 6-1Angles of Polygons
Tuesday, April 10, 2012
Essential Questions
How do you find and use the sum of the measures of the interior angles of a polygon?
How do you find and use the sum of the measures of the exterior angles of a polygon?
Tuesday, April 10, 2012
Vocabulary
1. Diagonal:
Tuesday, April 10, 2012
Vocabulary
1. Diagonal: A segment in a polygon that connects a vertex with another vertex that is nonconsecutive
Tuesday, April 10, 2012
Theorems
6.1 - Polygon Interior Angles Sum:
6.2 - Polygon Exterior Angles Sum:
Tuesday, April 10, 2012
Theorems
6.1 - Polygon Interior Angles Sum: The sum of the interior angle measures of a convex polygon with n sides is found with the formula
S = (n−2)180
6.2 - Polygon Exterior Angles Sum:
Tuesday, April 10, 2012
Theorems
6.1 - Polygon Interior Angles Sum: The sum of the interior angle measures of a convex polygon with n sides is found with the formula
S = (n−2)180
6.2 - Polygon Exterior Angles Sum: The sum of the exterior angle measures of a convex polygon, one at each vertex, is 360°
Tuesday, April 10, 2012
Polygons and Sides
Tuesday, April 10, 2012
Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Tuesday, April 10, 2012
Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Tuesday, April 10, 2012
Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Tuesday, April 10, 2012
Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 8 9 10 12 n
Name Octagon Nonagon Decagon Dodecagon n-gon
Tuesday, April 10, 2012
Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 8 9 10 12 n
Name Octagon Nonagon Decagon Dodecagon n-gon
Tuesday, April 10, 2012
Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 8 9 10 12 n
Name Octagon Nonagon Decagon Dodecagon n-gon
Tuesday, April 10, 2012
Example 1Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
Tuesday, April 10, 2012
Example 1Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S = (n−2)180
Tuesday, April 10, 2012
Example 1Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S = (n−2)180
S = (9−2)180
Tuesday, April 10, 2012
Example 1Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S = (n−2)180
S = (9−2)180
= (7)180
Tuesday, April 10, 2012
Example 1Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S = (n−2)180
S = (9−2)180
= (7)180
=1260°
Tuesday, April 10, 2012
Example 1Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S = (n−2)180
S = (9−2)180
= (7)180
=1260°
S = (n−2)180
Tuesday, April 10, 2012
Example 1Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S = (n−2)180
S = (9−2)180
= (7)180
=1260°
S = (n−2)180
S = (17−2)180
Tuesday, April 10, 2012
Example 1Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S = (n−2)180
S = (9−2)180
= (7)180
=1260°
S = (n−2)180
S = (17−2)180
= (15)180
Tuesday, April 10, 2012
Example 1Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S = (n−2)180
S = (9−2)180
= (7)180
=1260°
S = (n−2)180
S = (17−2)180
= (15)180
=2700°
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
11x +4+5x +11x +4+5x =360
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
11x +4+5x +11x +4+5x =360 32x +8=360
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
11x +4+5x +11x +4+5x =360 32x +8=360
−8 −8
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
11x +4+5x +11x +4+5x =360 32x +8=360
−8 −8 32x =352
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
11x +4+5x +11x +4+5x =360 32x +8=360
−8 −8 32x =352 32 32
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
11x +4+5x +11x +4+5x =360 32x +8=360
−8 −8 32x =352 32 32
x =11Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
11x +4+5x +11x +4+5x =360 32x +8=360
−8 −8 32x =352 32 32
x =11
m∠R = m∠T =5(11)
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
11x +4+5x +11x +4+5x =360 32x +8=360
−8 −8 32x =352 32 32
x =11
m∠R = m∠T =5(11) =55°
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
11x +4+5x +11x +4+5x =360 32x +8=360
−8 −8 32x =352 32 32
x =11
m∠R = m∠T =5(11) =55°
m∠S = m∠U =11(11)+4
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
11x +4+5x +11x +4+5x =360 32x +8=360
−8 −8 32x =352 32 32
x =11
m∠R = m∠T =5(11) =55°
m∠S = m∠U =11(11)+4 =121+4
Tuesday, April 10, 2012
Example 2Find the measure of each interior angle of parallelogram RSTU.
S = (n−2)180
S = (4−2)180
= (2)180 =360°
11x +4+5x +11x +4+5x =360 32x +8=360
−8 −8 32x =352 32 32
x =11
m∠R = m∠T =5(11) =55°
m∠S = m∠U =11(11)+4 =121+4 =125°
Tuesday, April 10, 2012
Example 3Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior angles of the octagon.
http://www.parkcitycenter.com/directory
Tuesday, April 10, 2012
Example 3Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior angles of the octagon.
http://www.parkcitycenter.com/directory
S = (n−2)180
Tuesday, April 10, 2012
Example 3Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior angles of the octagon.
http://www.parkcitycenter.com/directory
S = (n−2)180
S = (8−2)180
Tuesday, April 10, 2012
Example 3Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior angles of the octagon.
http://www.parkcitycenter.com/directory
S = (n−2)180
S = (8−2)180
= (6)180
Tuesday, April 10, 2012
Example 3Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior angles of the octagon.
http://www.parkcitycenter.com/directory
S = (n−2)180
S = (8−2)180
= (6)180
=1080°
Tuesday, April 10, 2012
Example 3Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior angles of the octagon.
http://www.parkcitycenter.com/directory
S = (n−2)180
S = (8−2)180
= (6)180
=1080° 1080°
8
Tuesday, April 10, 2012
Example 3Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior angles of the octagon.
http://www.parkcitycenter.com/directory
S = (n−2)180
S = (8−2)180
= (6)180
=1080° 1080°
8 =135°
Tuesday, April 10, 2012
Example 4The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
Tuesday, April 10, 2012
Example 4The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n = (n−2)180
Tuesday, April 10, 2012
Example 4The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n = (n−2)180
150n =180n−360
Tuesday, April 10, 2012
Example 4The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n = (n−2)180
150n =180n−360 −180n −180n
Tuesday, April 10, 2012
Example 4The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n = (n−2)180
150n =180n−360 −180n −180n
−30n = −360
Tuesday, April 10, 2012
Example 4The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n = (n−2)180
150n =180n−360 −180n −180n
−30n = −360 −30 −30
Tuesday, April 10, 2012
Example 4The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n = (n−2)180
150n =180n−360 −180n −180n
−30n = −360 −30 −30
n =12
Tuesday, April 10, 2012
Example 4The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n = (n−2)180
150n =180n−360 −180n −180n
−30n = −360 −30 −30
n =12
There are 12 sides to the polygon
Tuesday, April 10, 2012
Example 5Find the value of x in the diagram.
m∠1=5x +5, m∠2=5x, m∠3= 4x −6, m∠4=5x −5, m∠5= 4x +3, m∠6=6x −12, m∠7=2x +3
Tuesday, April 10, 2012
Example 5Find the value of x in the diagram.
m∠1=5x +5, m∠2=5x, m∠3= 4x −6, m∠4=5x −5, m∠5= 4x +3, m∠6=6x −12, m∠7=2x +3
5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360
Tuesday, April 10, 2012
Example 5Find the value of x in the diagram.
m∠1=5x +5, m∠2=5x, m∠3= 4x −6, m∠4=5x −5, m∠5= 4x +3, m∠6=6x −12, m∠7=2x +3
5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360 31x −12=360
Tuesday, April 10, 2012
Example 5Find the value of x in the diagram.
m∠1=5x +5, m∠2=5x, m∠3= 4x −6, m∠4=5x −5, m∠5= 4x +3, m∠6=6x −12, m∠7=2x +3
5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360 31x −12=360
31x =372
Tuesday, April 10, 2012
Example 5Find the value of x in the diagram.
m∠1=5x +5, m∠2=5x, m∠3= 4x −6, m∠4=5x −5, m∠5= 4x +3, m∠6=6x −12, m∠7=2x +3
5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360 31x −12=360
31x =372 x =12
Tuesday, April 10, 2012
Check Your Understanding
Review #1-11 on p. 393
Tuesday, April 10, 2012
Problem Set
Tuesday, April 10, 2012
Problem Set
p. 394 #13-37 odd, 49, 59
"They can because they think they can." - VirgilTuesday, April 10, 2012