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Interior Angle Sums
GEOMETRY
NAME_________________________
Polygons
DATE __________ Per.___________
Draw
diagonals
from
the
hollow
point
to
each
vertex
of
the
polygon.
1.
a)
b)
c)
d)
e)
2.
3.
polygon
# of sides
triangle
3
a)
tetragon
b)
pentagon
c)
hexagon
d)
septagon
e)
octagon
# of interior ∆’s
sum of interior ∠’s
1
What is the formula for the interior angle sum of a polygon with n sides?
Draw diagonals from the hollow point to each vertex of the polygon.
a)
b)
c)
180°
interior ∠ sum
=
d)
e) In each polygon, what is the sum of the angles’ degrees whose vertices are on the hollow point?
polygon
a)
triangle
b)
square
c)
hexagon
d)
octagon
# of sides
3
# of interior
∆’s
3
sum of all ∠’s
3(180°) = 540°
-m∠’s around
=
interior
∠ sum
540°– 360° =180°
___°– 360° = ___°
1080°– ___° =
8(180°) =
4.
What formula would you derive for the interior
angle sum using the method in #3?
5.
Are the formulas in #2 and #4 equal? If so, by what property?
GEOMETRY
Interior Angle Measures NAME_________________________
DATE __________ Per.___________
Polygons
polygon is equilateral (has congruent sides) and equiangular (has congruent angles.) Complete
1. Atheregular
table below for each regular polygon.
polygon
a) triangle?
b) quadrilateral?
# of angles
3
interior ∠ sum
(3-2)180° = 180°
interior ∠ measure
180°/3 = 60°
360°/4 =
c) pentagon?
(5-2)180° =
d) hexagon?
e) octagon?
f) decagon?
(8-2)180°/8 =
g) dodecagon?
h) n-gon?
(8)180°=
12
nsides
2. If a polygon has n sides, how do you find its interior angle sum?
How do you find the interior angle measure of a regular n-gon?
m interior ∠ =
Write the general formula for the interior angle measure of an n-gon.
3.
These polygons are convex.
These polygons are concave.
Draw all possible diagonals for each set of polygons .
4.
Complete each definition for convex and concave polygons.
A polygon is convex if and only all its diagonals are always ________ the polygon.:
A polygon is concave if and only if at least one diagonal is ________ the polygon:
All of this polygon’s interior angles add up
5. a)
to 900°. How many sides does it have?
b) An enneagon has an interior angle measuring 140°.
How many sides does it have?
Exterior Angle Sums
GEOMETRY
NAME_________________________
Polygons
DATE __________ Per.___________
∠2,
∠4,
and
∠6
are
exterior
angles
of
the
partial
polygon
pictured.
1.
Use the terms adjacent, supplementary, and linear pair towrite a definition for an
exterior angle of a polygon in a biconditional (if and only if) statement.
6
5
4
3
1 2
2.
Measure each exterior angle and calculate the sum for this quadrilateral.
Measure
each exterior
angle and calculate
the sum for this triangle.
3.
Sum of
exterior ∠’s =
Sum of
exterior ∠’s =
4.
Draw the remaining exterior angles for each regular polygon below.
Find the measure of each exterior angle (HINT: supplementary to interior angle).
Calculate the exterior angle sum.
a)
b)
Sum of triangle’s
exterior ∠’s =
c)
Sum of square’s
exterior ∠’s =
5. What would you guess is the exterior angle sum of any convex
Sum of hexagon’s
exterior ∠’s =
polygon?
Exterior Angles
GEOMETRY
NAME_________________________
DATE __________ Per.___________
Polygons
that each interior/exterior angle pair
6. Note
of a polygon is supplementary.
interior ∠
regular
polygon # of ∠ ‘s exterior ∠ measure
triangle
3
square
pentagon
hexagon
octagon
decagon
exterior ∠
Use that information to calculate the exterior
angle measures and sums of each regular
polygon in the table at right.
a) Use the table in #6 to graph the exterior
angle sums of an n-sided regular polygon.
7.
e
x
t
e
r
i
o
r
900°
720°
540°
360°
180°
S
u
m
0°
1
2
3
4
5
6
7
8
9
10
3(120°)=360°
4(
)=
Use the table in #6 to graph the exterior angle
8. a)
measures of an n-sided regular polygon.
y
E
x
t
e
r
i
o
r
180 - 60 = 120°
exterior
∠ sum
x
∠
y
120°
100°
80°
60°
40°
20°
0°
1
# of sides of a regular n-sided polygon
2
3
4
5
6
7
8
9
10
x
# of sides of a regular n-sided polygon
b) Are the graphed points collinear?
c) What does the graph imply for n = 7 & 9?
b) Are the graphed points collinear?
d) Find an equation for the line through the pts.:
c) Is each quotient an exterior angle measure?
360÷3 =
360÷4 =
360÷5 =
e) Does this graph verify your guess in #5?
f) Should these plotted points be connected?
Why not?
9.
360÷6 =
360÷8 =
360÷10 =
d) Find a equation for the curve through the pts.:
(HINT: See part c)
Recall that adjacent interior/exterior angles are linear pairs. So, in a polygon with n-sides, there are also
____ vertices. Hence, there are also n pairs of _____________ angles. The sum of all of the
polygon’s supplementary angles is______, which is the sum of the interior and exterior ∠’s. Substitute
the formulas below to derive the exterior angle sum of any convex polygon.
exterior ∠ sum = sum of supplementary∠’s – interior ∠ sum
formula
(
) – ______( ___ – ___ )
exterior ∠ sum= =
=
10.
Show that:
180 −
180(n − 2) 360
=
n
n
Interior Angle Graph
GEOMETRY
NAME_________________________
DATE __________ Per.___________
Polygons
1.
Plot the points for the measure of one interior angle of a regular n-gon.
The formula is . 180(n − 2)
n
i
n
t
e
r
i
o
r
200°
a
n
g
l
e
80°
180°
160°
140°
120°
100°
60°
40°
20°
0°
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
# of sides of a regular n-sided polygon
2.
Which regular polygons have integer interior angle measures?
3.
Are the points of this graph collinear or does it appear to level off? (If so, where does it level?)
4.
Name the regular polygon that has smallest possible interior angle measure. What is that
measure?
5.
A regular polygon with the largest possible integer measure for an interior angle has how many
sides? Use the formula or the Guess and Check method.
GEOMETRY Interior Angle Sum Graph NAME_________________________
DATE __________ Per.___________
Polygons
1.
Plot the points for the interior angle sums of n-sided polygons on the graph below.
Interior ∠ Sum = 180 (n – 2)
y
1400°
does it not make sense to use a
2. Why
continuous line for this graph?
1300°
1200°
i
n
t
e
r
i
o
r
a
n
g
l
e
s
u
m
3.
Should points be plotted for n = 1 or 2? Why
(not)?
4.
Why are no negative values graphed for the
x-axis?
5.
Why are no negative values graphed for the
y-axis?
6.
Are the graphed points collinear?
7.
Draw a line through these points.
a) Where does it cross the x-axis?
b) What does this tell you about a 2-sided
figure?
8.
What is the slope of this line? (Pay
9.
Find an equation in slope-intercept form y
= mx + b for the line.
1100°
1000°
900°
800°
700°
600°
500°
400°
attention to the scaling of the grid)
300°
200°
this verfiy the formula for interior
10. Does
angle sums, 180 (n–2)? Why?
100°
0°
1
2
3
4
5
6
n - # sides
7
8
9
10
x