geometry properties and attributes of polygons
DESCRIPTION
Geometry Properties and Attributes of Polygons. Warm up. Solve by factoring: 1) x 2 + 3x – 10 = 0 2) x 2 - x – 12 = 0 3) x 2 - 12x = - 35. Properties and Attributes of Polygons. Today you will learn about the parts of polygon and the ways to classify polygons. - PowerPoint PPT PresentationTRANSCRIPT
CONFIDENTIAL 1
Geometry
Properties and Attributes of Polygons
CONFIDENTIAL 2
Warm up
Solve by factoring:
1) x2 + 3x – 10 = 0
2) x2 - x – 12 = 0
3) x2 - 12x = - 35
CONFIDENTIAL 3
Today you will learn about the parts of polygon and the ways to classify polygons.
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 Attributes of Polygons
diagonal
A
E
B
CD
sidevertex
CONFIDENTIAL 4
You can name a polygon by the number of its sides. The table shows the names of some common polygons. Polygon ABCDE in the previous slide is a pentagon.
Number of Slides Name of Polygon
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
n n - gon
CONFIDENTIAL 5
Identifying Polygon
Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides:
PolygonPentagon
Not a PolygonPolygonOctagon
CONFIDENTIAL 6
Now you try!
Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides:
1a 1b 1c
CONFIDENTIAL 7
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.
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.
concave quadrilateral
convex quadrilateral
CONFIDENTIAL 8
Classifying Polygons
Next page ->
Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:
A
irregular convex
CONFIDENTIAL 9
Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:
B
regular convex
C
irregular concave
CONFIDENTIAL 10
Now you try!
Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:
2a 2b
CONFIDENTIAL 11
To find the sum of the interior angles measure 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 measures of the polygon.
Triangle
Quadrilateral
Pentagon Hexagon
CONFIDENTIAL 12
Polygon Number of Slides
Number of Triangles
Sum of Interior Angle Measures
Triangle 3 1 (1) 180° = 180°
Quadrilateral 4 2 (1) 180° = 360°
Pentagon 5 3 (1) 180° = 540°
Hexagon 6 4 (1) 180° = 720°
n - gon n n - 2 (n - 2) 180°
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°.
CONFIDENTIAL 13
The sum of the interior angle measures of a convex polygon with sides n is
(n - 2) 180°.
Polygon Angle Sum Theorem
CONFIDENTIAL 14
Finding Interior Angle Measures and Sums in Polygons
A) Find the sum of the interior angle measures of a convex octagon.
(n - 2) 180°
= (8 - 2) 180°
= 1080°
An octagon has 8 sides. So, substitute 8 for n.
Polygon ∕ Sum thm.
Simplify.
CONFIDENTIAL 15
Finding Interior Angle Measures and Sums in Polygons
B) Find the measure of each interior angle of a regular nonagon.
(n - 2) 180°
= (9 - 2) 180°
= 1260°
Substitute 9 for n.
Polygon ∕ Sum thm.
Simplify.
Step1: Find the sum of the interior angle measures.
Step2: Find the measure of one interior angle.
1260° = 140° 9
The int. ∕s are congruent, so divide by 9.
CONFIDENTIAL 16
Finding Interior Angle Measures and Sums in Polygons
C) Find the measure of each interior angle of a quadrilateral PQRS.
(4 - 2) 180° = 360°
m∕P + m∕Q + m∕R + m∕S = 360°
c + 3c + c + 3c = 360°
8c = 360° => c = 45°
Polygon ∕ Sum thm.
Polygon ∕ Sum thm.
Substitute.
P
Q R
Sc°
3c° c°
3c°
m∕P =m∕R = 45°
m∕Q = m∕S = 360°
CONFIDENTIAL 17
Now you try!
3a) Find the sum of the interior angle measures of a convex 15 - gon.
3b) Find the measure of each interior angle of a regular decagon.
CONFIDENTIAL 18
In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the
sum of the exterior angle measure is 360°.
81°
132°147°
147° + 81° + 132° = 360°
41°55°
110°43°
111°
43° + 111° + 41° + 55° + 110° = 360°
CONFIDENTIAL 19
The sum of the exterior angle measures , one angle at each vertex, of a convex polygon with sides n is
360°.
Polygon Exterior Angle Sum Theorem
CONFIDENTIAL 20
Finding Exterior Angle Measures in Polygons
A) Find the measure of each exterior angle of a regular hexagon.
A regular hexagon has 6 ext ∕s. So, divide the sum by 6.
Polygon ext ∕ Sum thm.
A hexagon has 6 sides and 6 vertices.
Sum of the exterior angle = 360°
Measure of one exterior angle = 360° = = 60° 6
The measure of each exterior angle of a regular hexagon = 60°
CONFIDENTIAL 21
Finding Exterior Angle Measures in Polygons
B) Find the value of a in polygon RSTUV.
Combine like terms.
Polygon ext ∕ Sum thm. 7a° + 2a° + 3a° + 6a° + 2a° = 360°
20a°= 360°
a = 60°
2a°3a°
6a°2a°
7a°
R
ST
U
V
So, divide the sum by 20.
CONFIDENTIAL 22
Now you try!
4a) Find the sum of the measures of exterior angle of a regular dodecagon.
4b) Find the value of r in polygon JKLM.
J
K
L
M
4r°7r°
5r°8r°
CONFIDENTIAL 23
Photography ApplicationThe appearance of the camera is formed by ten blades. The blades overlap to form a regular decagon. What is
the measure of ∕CBD?
m ∕CBD = 360° =36° 10
A regular decagon has 10 congruent ext. angles. So, divide the sum by 10.
∕CBD is an exterior angle of a regular decagon. By the polygon exterior angle sum theorem, the sum of the exterior
measures is 360°. B
D
A
C
CONFIDENTIAL 24
Now you try!
5) Suppose the shutter of the camera were formed by 8 blades. What would the measure of each exterior
angle be?
CONFIDENTIAL 25
Now some problems for you to practice !
CONFIDENTIAL 26
Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides:
Assessment
1) 2)
CONFIDENTIAL 27
3) 4)
Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:
CONFIDENTIAL 28
5) Find the measure of each interior angle of pentagon ABCDE.
A
B
C
D
E3x°
3x°
4x°
5x°
5x°
6) Find the measure of each interior angle of a regular dodecagon.
CONFIDENTIAL 29
7) Find the value of y in polygon JKLM.
8) Find the measure of each exterior angle of a regular pentagon.
J
K
L
M
4y°
4y° 2y°
6y°
CONFIDENTIAL 30
9) Name the polygon by the number of its sides.
10) In the polygon, /P, /R and /T are right angles and /Q is congruent to /S. What are m/Q and m/S?
R
P
Q S
T
CONFIDENTIAL 31
Today you will learn about the parts of polygon and the ways to classify polygons.
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 Attributes of Polygons
diagonal
A
E
B
CD
sidevertex
Let’s review
CONFIDENTIAL 32
You can name a polygon by the number of its sides. The table shows the names of some common polygons. Polygon ABCDE in the previous slide is a pentagon.
Number of Slides Name of Polygon
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
n n - gon
CONFIDENTIAL 33
Identifying Polygon
Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides:
PolygonPentagon
Not a PolygonPolygonOctagon
CONFIDENTIAL 34
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.
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.
concave quadrilateral
convex quadrilateral
CONFIDENTIAL 35
The sum of the interior angle measures of a convex polygon with sides n is
(n - 2) 180°.
Polygon Angle Sum Theorem
CONFIDENTIAL 36
Finding Interior Angle Measures and Sums in Polygons
B) Find the measure of each interior angle of a regular nonagon.
(n - 2) 180°
= (9 - 2) 180°
= 1260°
Substitute 9 for n.
Polygon ∕ Sum thm.
Simplify.
Step1: Find the sum of the interior angle measures.
Step2: Find the measure of one interior angle.
1260° = 140° 9
The int. ∕s are congruent, so divide by 9.
CONFIDENTIAL 37
In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the
sum of the exterior angle measure is 360°.
81°
132°147°
147° + 81° + 132° = 360°
41°55°
110°43°
111°
43° + 111° + 41° + 55° + 110° = 360°
CONFIDENTIAL 38
The sum of the exterior angle measures , one angle at each vertex, of a convex polygon with sides n is
360°.
Polygon Exterior Angle Sum Theorem
CONFIDENTIAL 39
Photography ApplicationThe appearance of the camera is formed by ten blades. The blades overlap to form a regular decagon. What is
the measure of ∕CBD?
m ∕CBD = 360° =36° 10
A regular decagon has 10 congruent ext. angles. So, divide the sum by 10.
∕CBD is an exterior angle of a regular decagon. By the polygon exterior angle sum theorem, the sum of the exterior
measures is 360°. B
D
A
C
CONFIDENTIAL 40
You did a You did a great great job today!job today!