Download 6.1 Polygons - Montgomery County Schools

Bell Ringer: Name the angle
opposite Line Segment AB
C
B
A
1
Bell Ringer: Name the angle
opposite Line Segment AB
Angle C
C
B
A
2
6.1 Polygons
Geometry
Mr. Peebles
Fall 2012
Objectives:
• “I can classify polygons to find the
measures of the interior and exterior angles
of polygons.”
3.4 Homework
• Pages 150-153
• (1-6, 10-15, 21, 28, 42-44)
5
3.4 Homework
• Pages 150-153
1. Angle 1 = 30 Degrees
2. Angle 1 = 83.1 Degrees
3. Angle 1 = 90 Degrees
4. x = 70, y = 110, z = 30
5. x = 80, y = 80
6. c = 60
10-15 Will check
6
3.4 Homework
• Pages 150-153
21. x = 147, y = 33
28. Will Check
42. G
43. B
44. H
7
Q
VERTEX
R
SIDE
Definitions:
P
S
VERTEX
T
• Polygon—a plane figure that meets the following
conditions:
– It is formed by 3 or more segments called sides, such
that no two sides with a common endpoint are collinear.
– Each side intersects exactly two other sides, one at each
endpoint.
• Vertex – each endpoint of a side. Plural is
vertices. You can name a polygon by listing its
vertices consecutively. For instance, PQRST and
QPTSR are two correct names for the polygon
above.
Example 1: Identifying Polygons
• State whether the figure is
a polygon. If it is not,
explain why.
• Not D – has a side that
isn’t a segment – it’s an
arc.
• Not E– because two of the
sides intersect only one
other side.
• Not F because some of its
sides intersect more than
two sides/
A
C
B
F
E
D
Figures A, B, and C are
polygons.
Polygons are named by the number
of sides they have – MEMORIZE
Number of sides
Type of Polygon
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
Polygons are named by the number
of sides they have – MEMORIZE
Number of sides
Type of Polygon
8
Octagon
9
Nonagon
10
Decagon
12
Dodecagon
n
n-gon
Convex or concave?
• Convex if no line that
contains a side of the
polygon contains a point
in the interior of the
polygon.
• Concave or non-convex if
a line does contain a side
of the polygon containing
a point on the interior of
the polygon.
See how this crosses
a point on the inside?
Concave.
See how it doesn’t go on the
Inside-- convex
Convex or concave?
• Identify the polygon
and state whether it is
convex or concave.
A polygon is EQUILATERAL
If all of its sides are congruent.
A polygon is EQUIANGULAR
if all of its interior angles are
congruent.
A polygon is REGULAR if it is
equilateral and equiangular.
Identifying Regular Polygons
• Remember:
Equiangular &
equilateral
• Decide whether the
following polygons
are regular.
Equilateral, but not
equiangular, so it is
NOT a regular
polygon.
Heptagon is equilateral, but
not equiangular, so it is NOT
a regular polygon.
Pentagon is
equilateral and
equiangular, so it is
a regular polygon.
Theorem 6.1: Interior Angles of a
Quadrilateral
• The sum of the
measures of the
interior angles of a
quadrilateral is 360°.
2
3
1
m1 + m2 + m3 + m4 = 360°
4
Ex. 4: Interior Angles of a
P
Quadrilateral
• Find mQ and mR.
• Find the value of x. Use
the sum of the measures
of the interior angles to
write an equation
involving x. Then, solve
the equation. Substitute
to find the value of R.
80°
70°
x°
Q
2x°
R
S
P
Ex. 4: Interior Angles of a
Quadrilateral
x°
Q
x°+ 2x° + 70° + 80° = 360°
3x + 150 = 360
3x = 210
x = 70
80°
70°
S
2x°
R
Sum of the measures of int. s of a
quadrilateral is 360°
Combine like terms
Subtract 150 from each side.
Divide each side by 3.
Find m Q and mR.
mQ = x° = 70°
mR = 2x°= 140°
►So, mQ = 70° and mR = 140°
Ex. 5: Polygon Sum Angle Theorem
– Exterior Angles
The sum of the measures of the
exterior angles of a polygon is
360 degrees.
Ex. 6: Polygon Sum Angle Theorem
The sum of the measures of the
angles of an n-gon is (n-2)180
Find the sum of the measures of
the angles of a 12-gon
Ex. 6: Polygon Sum Angle Theorem
The sum of the measures of the
angles of an n-gon is (n-2)180
Find the sum of the measures of
the angles of a 12-gon
n = 12, so (12-2)180 =
10(180) = 1800
Ex. 7: Polygon Sum Angle Theorem
The sum of the measures of the
angles of an n-gon is (n-2)180
Find the sum of the measures of
the angles of a 15-gon
Ex. 7: Polygon Sum Angle Theorem
The sum of the measures of the
angles of an n-gon is (n-2)180
Find the sum of the measures of
the angles of a 15-gon
n = 15, so (15-2)180 =
13(180) = 2340
Assignment
• pp. 161, 164
• # 1- 4 all, 8-15 all, 71-73 All
Exit Quiz: Find the value of x.
Find x
65

x
(2x-16)
Angle 1
24