Notes on Angles & Polygons
Name_________________________
Sides
Type
Sides
Type
4
quadrilateral
8
octagon
5
pentagon
9
nonagon
6
hexagon
10
decagon
7
heptagon
n
n-gon
--------------------------------------------------------------------------------------------------------------------Let's see if you can recognize a pattern developing between the type of polygon and the sum of
the measures of its interior angles.
The sum of the measures of the interior angles of a triangle (3 sides) is _________.
The sum of the measures of the interior angles of a quadrilateral (4 sides) is _________.
The sum of the measures of the interior angles of a pentagon (5 sides) is __________.
--------------------------------------------------------------------------------------------------------------------B
Consider the pentagon pictured to the right. Draw all of the diagonals
that have point A as an endpoint.
A
C
How many triangles have been formed?
E
D
How could this information be used to figure out the sum of the interior
angles for the entire pentagon?
--------------------------------------------------------------------------------------------------------------------The sum of the measures of the interior angles for a polygon with n sides is (n  2)  180 .
The whole concept of this formula is this: the "n - 2" portion of the formula finds how many
triangles, drawn from the same vertex, exist inside of the polygon. When this is found, the
formula then multiplies the number of triangles by 180 degrees.
--------------------------------------------------------------------------------------------------------------------What is the sum of the measures of the interior angles of an octagon?
--------------------------------------------------------------------------------------------------------------------In the above problem, suppose you were told that each of the interior angles has exactly the same
measure. What would be the measure of each interior angle of that octagon?
--------------------------------------------------------------------------------------------------------------------A regular polygon is a polygon in which all sides are congruent and all angles are congruent.
What is the name for a regular triangle?
What is the name for a regular quadrilateral?
The measure of each interior angle of a regular n-gon is
(n  2)  180
.
n
As you can see, the numerator (top) of this formula is exactly the same as the previous formula.
In other words, you still need to find the sum of all of the interior angles. Once that is found, you
simply divide the total by the number of sides (n).
--------------------------------------------------------------------------------------------------------------------What is the measure of each interior angle in a regular 16-gon?
--------------------------------------------------------------------------------------------------------------------As with triangles, exterior angles can be drawn for any polygon by extending one of the sides.
In the figure below, an exterior angle (angle 1)
is formed by extending one of the sides of
the hexagon.
In the figure below, an exterior angle has
been drawn at every vertex.
2
3
1
1
4
6
5
--------------------------------------------------------------------------------------------------------------------If each of the exterior angles were cut out of the figure to the right above, and each of the
vertices of these angles were placed together, the angles would form a circle.
The sum of the measures of the exterior angles, one at each vertex,
for any convex polygon is ___________.
--------------------------------------------------------------------------------------------------------------------What is the sum of the measures of the exterior angles of a convex decagon?
--------------------------------------------------------------------------------------------------------------------In a regular polygon, if all of the interior angles are congruent, what do you think is true about all
of the exterior angles?
Therefore, the measure of each exterior angle for a regular polygon is
360
.
n
--------------------------------------------------------------------------------------------------------------------What is the measure of each of the the exterior angles of a regular nonagon?
Homework on Angles & Polygons
Name_________________________
For Questions 1-2, consider the polygon to the right.
1.
What type of polygon is pictured to the right?
2.
What is the sum of the measures of its interior angles?
--------------------------------------------------------------------------------------------------------------------3.
What is the name for a polygon with ten sides?
4.
What is the sum of the measures of its interior angles?
--------------------------------------------------------------------------------------------------------------------For Questions 5-10, provide the requested information for the given type of polygon. Round all
decimals to the nearest tenth. Assume all polygons are convex.
5. pentagon - sum of the measures of
the exterior angles, one at each vertex
6. nonagon - sum of the measures of
its interior angles
7. regular 20-gon - measure of each of
its interior angles
8. regular octagon - measure of each
of its exterior angles
9. regular heptagon - measure of each of
its interior angles
10. hexagon - sum of the measures
of its interior angles
--------------------------------------------------------------------------------------------------------------------11.
Consider a regular 18-gon.
A)What is the sum of the measures of its interior angles?
B)What is the measure of each of its interior angles?
C)What is the sum of the measures of its exterior angles, one at each vertex?
D)What is the measure of each exterior angle?
--------------------------------------------------------------------------------------------------------------------12.
A question asks, "What is the measure of each interior angle in a pentagon?" As it is
currently written, the question cannot be answered. Add one word to this question so
that it can be answered.
13.
What is the value of x in the figure to the left?
9x
11x
--------------------------------------------------------------------------------------------------------------------14.
The measure of each interior angle of a regular polygon is 140 . What type of polygon
is it? You may use "guess and check" or any other problem solving technique.
--------------------------------------------------------------------------------------------------------------------15.
A triangle is to have side lengths of 31 inches and 43 inches. It is required that each side
length be a whole number of inches.
A)What is the shortest possible length for the third side of the triangle?
B)What is the longest possible length for the third side of a triangle?
--------------------------------------------------------------------------------------------------------------------PQR is constructed such that PQ = 8 units, QR = 17 units, and RP = 21 units. List
16.
the three interior angles in order from the angle with the smallest measure to the angle
with the largest measure.
--------------------------------------------------------------------------------------------------------------------For Questions 17-18, find the requested value.
17. WXYZ is a parallelogram. If
mZ  6a  8 , and mY  3a 19 ,
then what the value of a?
18. KNJM is a kite. If NJ = 15,
and LJ = 9, what is LN?
W
N
Z
6a - 8
K
X
L
3a - 19
J
Y
M
--------------------------------------------------------------------------------------------------------------------19.
EFGH is a rectangle. If EF = 8, and FG = 15, what is EG?
--------------------------------------------------------------------------------------------------------------------20.
In isosceles trapezoid HIJK, HJ = 5x + 4 units, and IK = 6x - 8 units. Is there enough
information to find x? If so, what is x?