Download 6.1 Polygons

10.1 Polygons
Geometry
Objectives:
• Identify, name, and describe polygons
• Use the sum of the measures of the interior
and exterior angles.
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.
Interior angles of quadrilaterals
• The sum of the measures of
the interior angles of a
quadrilateral is 2(180°), or
360°.
B
C
A
B
D
C
A
A
C
D
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
80°
• 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.
70°
x°
2x°
Q
x°+ 2x° + 70° + 80° = 360°
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°
Practice Together
Practice Together
Practice Together
= 157.5
180 s – 360 = 157.5 s
22.5 s = 360
S = 16