Download Chapter 1 Tools Of Geometry

Drill
1) If two angles of a triangle have a
sum of 85 degrees find the third
angle.
2) The three angles of a triangle are 2x,
3x, and 2x + 40 find each angle.
2.2
Polygons
Polygon
Is a closed figure with at least three
sides, so that each segment intersects
exactly two segments at their
endpoints.
Polygon Terminology
Sides
Vertex
C
D
Interior
B
E
Diagonal
A
Consecutive
Vertices
F
Naming Polygons
A polygon can also be classified as convex or concave.
If all of the diagonals
lie in the interior of
the figure, then the
convex
polygon is ______.
If any part of a diagonal lies
outside of the figure, then the
concave
polygon is _______.
Types of Polygons
# of Sides
3
4
5
6
7
8
9
10
12
Name/Draw
TRIANGLE
QUADRILATERAL
PENTAGON
HEXAGON
HEPTAGON
OCTAGON
NONAGON
DECAGON
DODECAGON
Diagonals and Angle Measure
Make a table like the one below.
1) Draw a convex quadrilateral.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) 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 10-1
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.
Diagonals and Angle Measure
In §7.2 we identified exterior angles of triangles.
Likewise, you can extend the sides of any
convex polygon to form exterior angles.
57°
The figure suggests a method for finding the
sum of the measures of the exterior angles
of a convex polygon.
48°
72°
54°
When you extend n sides of a polygon,
n linear pairs of angles are formed.
The sum of the angle measures in each linear pair is 180.
sum of measure of
exterior angles
sum of measure of
exterior angles
=
sum of measures of
linear pairs
–
sum of measures of
interior angles
=
=
n•180
180n
–
–
180(n – 2)
180n + 360
=
360
74°
55°
Polygon Interior Angle-Sum Theorem
The sum of the measures of the interior
angles of an n-gon is (n-2)180.
Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior
angles of a polygon, one at each
vertex, is 360.
Homework
Pages 79 – 80
#’s 1 – 4, 10 - 26